Personally I'd suggest taking a look at differential backups.
You could do them at, say, 4 or 6 or 8 or 12 hour intervals between full backups, whatever works for you. They don't allow point-in-time, but would allow recovery to the last differential. Even a single differential 12 hours after each full backup would reduce loss to 12 hours max instead of 24. And diffs are generally small, so they don't require much resources to take or manage.
In fact, for most of my dbs, I only take full backups on weekends and do diffs during the week, to save time and resources on backups.
Transaction log backups are only possible if the db is not in simple mode. You need to verify the recovery model for the dbs too.
If you're not using the tran log backups, and don't plan to, you'd be far better off switching to simple recovery.
"Trashing about"? I don't follow. The log file is sequential, period.
Never set auto-shrink on, never, period.
I understood that, I knew you were talking about other IT people. I've run into many myself. Some even try to tell me I don't know what I'm talking about! Very frustrating.
Exact opposite. Full recovery does some logging that simple recovery does not do. That is, changing from simple to full will typically significantly increase the amount of logging that occurs. You need to make allowance for that overhead.
https://docs.microsoft.com/en-us/sql/relational-databases/logs/manage-the-size-of-the-transaction-log-file?view=sql-server-ver15