This is in addition to farcher's answer.
An analogy for The current in the circuit
A person is standing , and there is a very long multiple line of restless kids along with their parents, kids are moving around (due to hot weather) in front of a the person , and there is a Small stretch of Barricades in between the kids s somewhere in the the line. Something like this
(I forgot to draw the parents, assume they are there.)
Now the person(battery) shouts "snake" (creates electric field in the wire), the kids in front of him immediately hear it and starts running away from him ( effect of electric field on charges),
[but kids at the other end haven't stated to move as sound didn't reach them yet (speed of light is capped and not infinite, so is electric field)],
now moving kids stumble with other's parents and fall down , but get up and start running again this keeps repeating, (parents are smart won't run ; ) )
and region where the barricades(resistor)¹ are present , the kids stumble with barricades a lot more (high Collison frequency, and decrease in time between collisions), beacause of this the kids before the barricades experience traffic snake effect, aka traffic wave . [Video on it] (https://m.youtube.com/watch?v=Q78Kb4uLAdA)
So the all kids before the Barricades move slower than their max capability ( if barricades wasn't there at all). Till now we just saw what happens upto charge flow to a certain point in the circuit
Now comming back to real world
It's a fact that charge leaving battery is equal to charge entering the battery,
so when resistor is connected as explained above charge leaving the battery decreases
So consequently charge entering the battery decreases
, Therefore current decreases in the entire circuit.
So basically this explains the statement " resistor not only has an effect on the current passing through it but also the currents passing through entire circuit to which it is connected".
So statement in my question is disproved.
Also this an original analogy that I came up with, although not perfect , but it makes people visualise lots of things.
¹Please note that barricades in the resistor actually don't exist , the obstacles in the resistor is of similar size as that of condutor. Here it's said so to simplify the explanation.