Does over-discharging damage batteries?

Yes. For a 12 V VRLA, end-of-discharge voltage is typically around 9.5 to 10 V; taking it below that causes damage that is often permanent. The usual cause is not operator error but a system drawing more power than the battery was sized to deliver, a load that has grown since the bank was specified. If a bank is repeatedly going flat, the problem is the sizing, not the battery.