we distinguish between accidental damage, that I pay for, and wear and tear.
in the case...
the gate is old and falling apart. if you closed it in a way that would not cause damage to a normal gate, and it broke because it was in poor condition, then this is normal wear and tear.
if you tripped over (for example), fell on the gate and it broke, it is accidental damage.
insurance policies will usually pay for accidental damage but not wear and tear.
also if you rent our property, your tennants are responsible for accidental damage but you are responsible for normal wear and tear.
in your case it sounds like normal wear and tear that broke the gate (i.e., the years of not maintaining for it damage it, you were just the guy who happened to close it on the day it gave up). in your possiition i might offer to fix it (like you did) as a good will jesture. i would not however pay for a new gate.
a client who is reasonable will be OK with this distinction. (of course, some clients are not reasonable
)
hope this helps