It's not necessarily an issue of fuel, but the overall wear on the components and engine when you start a car. A starter motor only has so many "starts" it can do before dying. The battery too.
Starter motors have gotten a lot better since the "bad old days" and engines start more smoothly thanks to fuel injection and computer control systems, so manufacturers have decided that it's ok to start/stop engines as needed, but the reason for not doing it was never a matter of fuel savings.
If you're on Android, that's probably the most important thing to consider first. Just using a phone with Google Play Services gives up a great deal of privacy, even if you think you've turned all that off.
So, consider either an iPhone or if you're really serious about this shit, GrapheneOS. The iPhone is easier for average people, but GrapheneOS is more private and you have more control. Both are better than Googled Android.
For a Google Drive alternative, that's simple. If you're already using Proton, get ProtonDrive.
For Google Docs alternative, that's a bit more challenging. There aren't any direct competitors with full end-to-end encryption that are any good and also cross-platform.