Jumping on a bit late, but yes, typical thermal circuit breakers do become weaker over time. The internal parts wear from use and much like old springs in a car can no longer hold car at proper level the moving parts in the breaker can no longer handle as much current.
While you may or may not be able to find the exact same part, there are hundreds of choices out there for replacement breakers and it is likely you can find one that will fit in the same hole in the brick with little trouble. A quick look at Digikey shows breakers of a similar style and rating to start at about $2.60.
In the grand scheme of things, the Z-1000 is really just a mid-range transformer, not really intended for powering lots of incandescent lights or multi-motored post-war locomotives. Converting to LED lights will greatly decrease the power demands of your layout, but switching to a more powerful transformer may also be a good idea. Just don't think you need a monster ZW-L or Z4000 to replace a Z1000. Even the monster transformers only provide 10 Amps to a single block of track.
On a last note, I am also confused by this:
After about 20 seconds the breaker would reset itself and the train was good for another dozen laps or so.
Are there versions of the Z1K that have self resetting breakers? If not, it's possible someone has already had a go at repairing this transformer and may have installed the wrong current rating breaker. I can't say with certainty that my z1000 is correct, as I've found out it came from the factory wired backwards of how most of them are, but mine has a standard, run of the mill, 6 Amp thermal breaker that has to be pushed to be reset.
JGL