Common Cycle in Ann Arbor, MI was used Google sheets as you described.  We also explored the use of TimeCounts: https://timecounts.org/

I can't comment on if they are still using TC.  It has a lot of benefits, and acts like a CRM.  I recommend it if you're really committed to diving into a more active relationship with your data and volunteer contacts.  The downside for your use is that it's not as easy to pop open the sheet and check availability, because each user needs a login.  That's compared to Google, where most users are already logged in to a compatible email account, or you can simply allow public access.

Bikes Together in Denver, CO uses MyVolunteerPage from Better Impact.  I only have experience with this on the volunteer end, not the admin side.  It's a good tool because it allows on-site timesheet check-in, and sign-up ahead of time.  Based on my experience, volunteers are good about using the check-in feature, but the sign-up feature doesn't seem to get a lot of use.  Most people seem to depend on the same crew being available most weekends, and those folks tend to communicate off-line about any gaps in their availability.  Overall, these factors are cultural, so your mileage with any given software may vary.

Andrew