This question highlights an important point of confusion in many explanations of clock synchronization in frames of reference in special relativity: the sloppy use of the term "observer".
In actual fact, many explanations forget to explicitly highlight that an admissible special relativistic frame of reference, when imaginged as a lattice of clocks, not only has to synchronize its clocks but it also comes equipped with a little assistant at each synchronized clock, each assistant only recording the events that occur in the immediate vicinity of her clock by reading and co-recording the (ever changing) local time of her clock and the (unchanging) local coordinates of her lattice point at her home position when the event occurs.
The "observer" is a ficticious entity that has access to all the notebooks of all the assistants. In standard physics usage an "observer" in special relativity is, for all practical purposes, simply equivalent to a frame of reference and its coordinate system's encoding of events. This coordinate system in turn is best imagined as the lattice of clocks and their respective local assistants or recording devices for events. If one were to insist that the observer were indeed real and sitting at the origin then she would have to wait until the updates of all the assistants' new entries in their respective notebooks have arrived at the origin (and even when being sent by a radio wave travelling at the speed of light this may take some time...). In theoretical physics an observer in special relativity is thus not much more than a coordinate frame and its events, and in experimental physics you'd better automate all the assistants' work (local particle detectors) and then evaluate all the recorded events off-line (see e.g. experiments at CERN).
This same procedure of establishing a lattice, synchronizing clocks, and putting a recording device for local events (assistant) at the position of each lattice point and its corresponding clock is executed for all admissible frames of reference (by admissible I mean inertial systems in which a light beam may be considered to follow a straight line).
Thereby even statements about simultaneity in a single frame of reference already involve multiple (at least two) assistants at different positions recording in their respective notebooks their local time read from their respective local clock, and we mean by the assistants doing this at a single point in time that these assistants happen to write down the same numbers, e.g. 12:00 at their different locations into their different notebooks of events.
So, for example, when measuring the length of a "rigid" rod that is moving at a constant velocity in a frame of reference 1, then one has to understand that also in the frame of reference 2 that is comoving with the rod (i.e. in which the rod is not moving at all) one has two assistants: one at each end of the rod. Or to put it differently: all the assistants of the comoving frame are instructed to check if they are at the rear end or the front end when their respective clock shows 12:00. This is an instruction that does not require the assistants to travel. Only to check events in their vicinity. If they are at one of these locations when their clock reads 12:00, they should mark their coordinate lattice reading into their notebooks. Afterwards, these entries in the different notebooks must be evaluated: there will be one notebook noting "rear end at 12:00 at position P1=(x1,y1,z1)" and another notebook noting "front end at 12:00 at position P2=(x2,y2,z2)". All the other notebooks will have no entries since neither the rear end nor the front end happened to be at any other locations at 12:00. The "observer", or the master of all assistants, or simply the physicist using the coordinates of the comoving frame of reference, will then get the length of the rod by computing the Euclidean distance between the spatial locations P1 and P2 in frame 2. This gives the length of the rod at rest, i.e. the length in the co-moving frame 2. The same procedure applies of course in coordinate frame 1, in which the rod is moving. Only in this frame one uses the synchronized clocks of frame 1 and its lattice and its assistants. One imagines that such different frames of reference do not physically disturb each other. In frame 1, the assistants will perform the same task, they too will note at their pre-specified local times, e.g. 12:00, if they happen to be at the rear end or at the front end and they will record their positions Q1 and Q2 in the local notebooks in frame 1. It is most important to realise that when the synchronized clocks of frame 1 show the same time 12:00 the corresponding clocks of frame 2 at the same physical positions as their counterparts in frame 1 would not show the time of frame 1, and would not even show identical times in frame 2 either. Hence, from the point of view of the comoving frame 2, the positions of the rear end and the front end of the rod are recorded by assistants in frame 1 at different comoving times in frame 2, even though the clocks of frame 1 would all show 12:00 and the procedure in frame 1 is perfectly sound: record the positons of both ends at the same time in frame 1. This is the origin of length contraction in special relativity. It is important to understand that this length contraction result is not what anyone "sees" with their eyes but rather the result of the entries in the notebooks of the assistants in frames 1 and 2. This result is generic and applies also to the original question: a single person's visual experience of possibly remote events is not what is usually meant by observations made by an "observer" in special relativity (note that e.g. observations in astrophysics usually already involve general relativity, see also the brief remarks on clocks in general relativity below. In any case also for these observations we also need to imagine little local assistants throughout the universe and then compute from their imagined detector readings what an astrophysicist on earth should in the end actually "see" in her telescope, e.g. red-shifts, the measurement event here being the impinging of a propagating light ray on the telescope's imaging plane and the recording of this event by the local astrophysicist/assistant).
A similar argument applies to a moving clock and ticks on this clock. The clock would be moving by the lattice of clocks of frame 1. And when one tick has passed for the moving clock, the clock of frame 1 which happens to be next to the moving clock would not show the same time as the moving clock, even if the moving clock and another clock of frame 1 had shown the same time at the beginning of the experiment when these two clocks were co-located. This is the origin of time dilation. Again it is important to understand that this is the result of the entries of events in the notebooks of the assistants in frames 1 and 2.
This viewpoint of local assistants only recording their local events and an imagined "global observer" evaluating all recordings is especially useful when it comes to generalize to general relativity: all you have to imagine then is that the latticework is no longer rigid but rather turns into a freely floating cloud of tiny dust particles (or "reference-mollusc" as Einstein put it) with each dust particle carrying its local coordinates as three numbers written onto it, and a local clock with a local assistant and her local notebook to record only events occurring in the vicinity of the assistant's respective dust particle (and assuming that all these imagined infinitesimal dust particles do not modify the background metric, which is akin to assuming that a charged infinitesimal test particle would not influence the local electric field it is meant to measure). The clocks can no longer be synchronized globally in general relativity but that is not necessary to log local events in the local notebook together with a local point in time and a local spatial coordinate, i.e. the reading of the unsynchronized clock and three numbers written on the local dust particle when and where the event occurred. Again the "observer" is an imagined physicist evaluating all the logs in all the notebooks (NB1: strictly speaking the most general coordinate system in general relativity would not even have three constant spatial locations written on the dust particle but rather only three numbers that may also change over time, e.g. imagine four clocks on a dust particle, or alternatively a smartphone showing four continuously changing numbers. The only really important condition is that the four clocks are continuously changing times without ever showing equal four numbers for different events, e.g. for different dust particles, and usually the mathematical condition that the coordinates should be smoothly changing from particle to particle is also desirable. The readings of these four clocks, or, equivalently, the four changing numbers on the smartphone display of the local assistant serve to specify the coordinates of an event taking place at the dust particle and recorded by the local assistant or recording device. NB2: as an aside, the fact that a freely falling local observer "in an elevator" can also establish an extremely tiny, in both space and time localized latticework of rigid rods and synchronized clocks, just like in special relativity, allows for measuring the general relativistic metric by comparison of the local special relativistic coordinates with their corresponding dust particle coordinates)
For further reading along these lines (assistants recording events in notebooks) I would recommend:
(1) "The Geometry of Minkowski Spacetime" by Gregory L. Naber, 2012, Springer, in particular the Introduction.
(2) "The Einstein Theory of Relativity: A Trip to the Fourth Dimension", by Lillian Lieber, Paul Dry Books, 2008, for an extremely easy introduction to both special and general relativity.
(3) "General Relativity from A to B" by Robert Geroch, The University of Chicago Press, 1978, for an equally easy introduction to general relativity.
(4) "Relativity: The Special and the General Theory", Albert Einstein, annotated 100th Anniversary Edition by Princeton University Press, 2019, discussing in chapter 28 the "reference-mollusc" as a pedagogical aid invented by the master himself.
(5) https://en.wikipedia.org/wiki/Length_contraction#Visual_effects for further references explaining the difference between what a single person would "see" and what is meant by measuring a length contraction.