Have fundamental particles been observed? Addendum: the answer appears to be either "no" or "depends on what you mean". Most of the "depends" involve a meaning of "particle" that is clearly jargon. My question was motivated by meeting people who claimed it was not jargon but an essentially lumpish quasi classical particle. I selected the answer of ACuriousMind because it was most on topic and most informative and because I think that this has proved to be a controversial matter - it was not my interest to be the center of a controversy.

In an answer to this question from about 5 years ago,
Do particles exist according to quantum field theory?
the assertion was made that whatever the argument about fields and particles are - we observe particles. But, if by particles is meant an infinitesimal hard lump, how can they ever be observed? Average field properties can be observed. The Born interpretation can be used as well as path integrals, but none of those are experimental evidence that on observing a quantum system somehow a particle appears for a moment and then vanishes again.
Ethan Siegel on Facebook (FWIW) claims that deep inelastic scattering experiments in the 1970s prove that particles exist. But, again, that is just average properties of fields. Again, it seems to be just a matter of interpretation to see fundamental particle scattering to be small hard lumps bouncing off each other.
The core of my question is - what is the status of the assertion that particles actually exist in the current thinking in physics?
See also this related question:
Must Matter Particles Have A Hard Edge?

@ACuriousMind asked me to clarify "exists". I personally have no idea what the word means in a physical sense - this is a word used by others.
I only use it with a clear meaning in terms of being an implication of a mathematical theory - there exists a solution to an equation, that sort of thing. So, in the answer, I would ask that other people clarify what they mean by particle and by exists - as I am looking for others to clarify something that makes no sense to me in the theory of QM. What I say is that the Modern mathematics of the Standard Model does not imply little hard lumps. I interpret that as a sense of "particles do not exist in QM". Of course, there are other definitions of particles.
But, I am interested in the concept little hard lump particles that suddenly appear when you make an observation of a quantum system. I have yet to see anything other than -- localized quanta of energy, etc. If this is just terminology, it is terminology that quite a few people seem to think is vitally important.
Beyond this, we descend into semantics. Which is something I personally eschew.
 A: I started my answer to the linked question by saying:

I think this is largely a matter of terminology.

and that remains the case. Take for example a hydrogen atom decaying to emit a photon. No physicist I know would doubt that here we observe a particle being created and when it hits our detector we observe that same particle being destroyed. If you are uneasy with the example of a massless particle you could instead use an example like a muon decaying to an electron and two neutrinos.
In QFT we usually consider the particle as an excitation of a Fock state, but the Fock state represents an infinitely delocalised particle i.e. a particle with a perfectly defined momentum but an infinite uncertainty on position. Obviously no such particles exist, so instead the objects we call particles exist as wavepackets that are superpositions of plane waves. That is, they are localised in space to within some distance $\Delta x$ and have a correspondingly uncertain momentum $\Delta p$.
When we detect the particle the measurement process collapses the particle wavefunction to a state with a much smaller position uncertainty and that produces the observation of the small heavy object.
A: 
Have fundamental particles been observed ?

As you say, we do not observe particles. However, we do observe phenomena (I am deliberately avoiding the word "object") with certain behaviours. These phenomena sometimes behave as a hard lump of something - which we call a particle - and at other times behave in a more diffuse way - which we call a wave. This is wave-particle duality. Because these phenomena appear to be indivisible (although they can transform into other phenomena) we call them "fundamental". We happen to call these wave/particle phenomena "fundamental particles", but we could just as easily have called them "fundamental waves" or "fundamental wave/particles". We give these phenomena different labels such as quark, electron, photon depending on their observed properties of mass, charge etc.
Sometimes it is simpler and more convenient to treat these phenomena as if they were particles. Sometimes it is simpler and more convenient to treat them as if they were waves. But both "particle" and "wave" are analogies that we have co-opted from our everyday experience of macroscopic phenomena, and neither analogy captures the complete quantum nature of these phenomena.
A: The universe exists. Physics is a tool to model the universe.
There are two parts to physics. The important part is the math. We say a theory is wrong if the math doesn't match the behavior of the universe. We may use it anyway because it is a good approximation in certain regimes. Classical physics is the easy example.
Then there is the mental model we use to guide our thinking. We are a lot looser about this. We know what we are describing is neither a wave nor a particle, but we use both concepts anyway.
It doesn't have to be totally right to be useful. Virtual particles are really perturbations. Add up all the terms described by a Feynman diagram if you want the true behavior. But each term can be thought of as a virtual particle that sort of exists.
In the same way, aether was a useful idea in its time. It "explained" how electromagnetic waves could exist as vibrations when there didn't appear to be anything to vibrate. The theory wasn't abandoned when aether needed to have counter-intuitive properties like permeating everything. It was dropped when the Michelson-Morley experiment didn't match reality.
So do fundamental particles exist? No. They are mental models to guide our thinking. Something exists that is being modeled. But it isn't exactly like a fundamental particle. It isn't a classical point or hard sphere.
Have they been observed? Yes. We have seen the behavior that gave rise to the idea. The behavior is close enough to particle-like in some circumstances that the idea is useful.
Ideas like particle and wave are questioned when they guide your thinking in wrong directions. This happens a lot, so they get questioned a lot. But they do help, so they get used a lot.
A: The notion of "particle" is a slippery one, and in different contexts this word means different things.
In the context of fundamental particles in QFT, the definition of a "fundamental particle" is that it is point-like, i.e. has no substructure. But in quantum field theory, these words don't actually refer to the classical idea of some sort of "hard lump", but they are technical claims about how the quantum state we call a "particle" behaves during scattering. For more discussion of the notion of point-like particles see Do electrons have shape? and What is the meaning of the size of a particle in QFT?.
Another aspect of the notion of "particle" arises when you fire particles into a bubble chamber, where we then observe trajectories that certainly look like some sort of classical "hard lump" leaving a trace in the medium of the chamber. However, this does not imply the hard ontological claim that there really "is" such a lump, since the emergence of these classical trajectories can be explained by the localized interaction of quantum states, see this excellent answer by rob.
So when you hear quantum physicists say "we observe particles", they usually mean it in the above sense: These quantum states have no discernible substructure, and when observed in environments like a bubble chamber, they look like the classical idea of a little corpuscular, even though we do not need to assume that there "is" such a corpuscular to explain the trajectory. Whether that makes the claim justified or whether you think this is a confusing abuse of language that hides the fundamentally quantum nature of these states is left to the reader to decide.
