It's not an axiom in the mathematical sense, so no, that argument alone is not a reason to just state it is true.
It's a physical law, and as such it's just a theory. But it has been confirmed by a lot of experiments, so lets assume for the moment it's true


Afterwards, the speed of the insect = the *new* speed of the train = the former speed of the train - some tiny value (ignoring friction and engine forces).


The momentum of the insect is zero, yes. But you always have to look at the momentum of the whole system. Momentum must be preserved, but it can be transferred from one object to another.
momentum_before = m(insect) x (speed(insect)=0) + m(train) x speed(train) = m(train) x speed(train)
That's the initial momentum of the whole system.


momentum_afterwards = m(insect) x speed(after) + m(train) x speed(after) = (m(insect)+m(train)) x speed(after) = m(train) x speed(train)
speed(after) = speed(train) x m(train) / (m(insect)+m(train))

So speed(after) is lower than speed(train) and higher than speed(insect)=0 (assuming m(insect) is finite and >0). The speed of any object must be continuous, because only finite acceleration is possible. And there is no force present that accelerates the train, so the curve of speed over time is monotonous. That means the speed of the train at any time is between speed(train) and speed(insect). Therefore, it is at no time equal 0.

The increased pressure inside the can blows it up a bit. In order to change the turning speed, which is required for rolling down a ramp, you need to apply torque, and that is proportional to the radius. You thus need more force to get the bigger can turning, which means that less force is available for acceleration downwards. The blown-up can therefore is slower in accelerating and arrives later at the bottom.

Starting out from my equations, we get that the insect accelerates from a negative velocity to a positive velocity, so obviously at some point after the collision it has to have velocity 0. The train decelerates from a large positive velocity to a slightly smaller positive velocity, so at no point it reaches zero. That's a direct consequence of the law of conservation of momentum.

The apparent paradox is that from the point on when they collide, they should have the same velocity. So how can it be that the insect will have 0 velocity at some point after the collision, but the train not?
The problem here is that it's not really correct that both have the same velocity. Their velocities will approach each other, but not instantly, because of deformation of the colliding bodies.

Flowovermind is the first one to state .

You are the first one to mention .

Now also flowovermind mentioned 'deformation'.

The bold terms in your comment above 'spherical', 'uniform density' - what do they have to do with the answer?

Why 'Therefore'?

If it could have happened in a single instant, would then the train stand still?

At the time when they are approaching zero speed difference,
the energy transfer by the momentum of the insect will also be approaching zero.
I think that's the answer. I'm not a physicist myself and heard this problem from a informatician 10 years ago, hence for me it was quite an honest puzzle.

It could be called Zeno and the Mosquito but this would probably give something away already.

Get your infinite go-board and choose Tengen. For each point p define n(p) as the (least) number of steps to get to Tengen. Define a potential V(p) = 3^-n(p). Check that at each p ( apart from Tengen ) the potential is equal to the average at its neighbours and that the potential drops to zero far away. Apart from Tengen for each point the net incoming current equals the net outgoing current. We have a solution for the case that one of Herman's points is kept at 1 Volt and infinity at 0 Volt.
Herman's other point will turn out to be at 1/27 Volt. Tengen's neighbourpoints will be at 1/3 V each. They will receive (1 - 1/3)/1 Ampere each from Tengen. In total Tengen radiates 8/3 Ampere. Next get your other infinite goboard out of your cupboard and choose Tengen again and put it at -1 Volt keeping infinity at zero. Next make a sandwich of them but choose the Tengens not to coincide but to correspond with Herman's points. Now we add the potentials. The first Tengen will be at 1 - 1/27 Volt = 26/27 Volt. The other at -26/27 Volt. A potentialdifference of 52/27 V. It (still) gives rise to a current of 8/3 Ampere. ( check yourself ) The resistence is (52/27)/(8/3) = 13/18 Ohm.

Pooh

4/pi - 1/2 ~= 0.77 Ohm