I found a solid metal strainer, not mesh for my sink. Holds up forever and much easier to clean. Even if you have a disposal, its not great to just dump everything down the drain [citation needed]. You especially want to catch things like small bones and forks.
The disposals seem common enough, a lot older/outdated homes don't have them. I have no idea what they're putting into new houses and renovations.
Never had an under the sink mess like you describe, maybe yours just had a small leak?
the 3rd road; Be awkward by making random noises.