How does transistor bandwidth affect stability?
You need to think of an amplifier like a control system. Imagine that you are trying to rotate a piece of machinery to a certain angle and using feedback from an angular position sensor to drive the error out of the system.

If the machinery is lightweight, it requires less energy to get it rotating and conversely, it requires less braking energy to stop it once it has reached it's "destination". So there may be a little bit of overshoot; the machine slightly overshoots the target position but the control loop brings it back quickly and the error is removed.

Now if that machine were much heavier, it would take more energy to move it and more energy to stop it and that likely means a greater overshoot past the target position and it could lead to sustained oscillation at the desired position. That oscillation frequency would be lower than the oscillation frequency of a lighter machine should it have similar problems.

So, faster transistors are like lighter machinery and slower transistors are like heavier machinery.

You can't rule out that a faster transistor might produce nuances not seen in the slower transistor but the chances are they are at a much higher operating frequency and easier to get rid of (or even ignore).