First computers used relays, becase they were already faster and cheaper than purely mechanical computer. Then came lamps which were even faster. Nowadays mechanical computers are just uneconomical curiosity, but thanks to miniturisation they look like they might be economical for several VERY special purposes where silicon just doesn't work.

> Was it because the mechanical parts are too prohibitively rigid/inflexible/heavy/expensive for a computer?

Yes, and it was all those reasons at once. Electrical/electronical computers were better at every metric simultaneously.