Are there any drawbacks to using a BaaS (backup as a service) provider? What are they?

Cost would be the only major consideration here, but this would all have to depend on the data that is being backed up. BAAS is a great service option, esp. considering the storage guarantees and knowing someone is monitoring the backup processes. Whilst i hate admitting it, as a CTO in my career we have been caught out a few times where backup processes hadn't worked for a week or so before someone caught it. Luckily we were able to just fix the issue and keep moving but lesson learnt!!! Would have been a catastrophe if we had an event!

BAAS needs compliance approval in controlled environment 

Upfront costs would be the biggest drawback. Also, training your team on how the service works takes time. Backups are a business critical application, so spending the time to make sure everything works and is tested. It is also crucial to make sure your team understands and practices the restoration process in the event of a future need.

Most BaaS platforms are costly upfront, but overtime they should become a value-add. 

Call me old school, but physical access to your backups, is still a good idea, whenever reasonable; at the very least, spread securely around to different geo-diverse locations, under your control.