Completely disabling cellular data and limiting apps only to WiFi when available
First, "not possible" is a straight lie. The carrier is perfectly capable of turning that feature off and limit you to "calls and SMS only". They simply don't want to.

Second, you could try some Firewall app in whitelist mode (i.e. by default all apps are blacklisted). Take e.g. a look at NetGuard (I recommend taking it from F-Droid, though it's also available at Playstore). This firewall app doesn't require rooting your device, and you can set up mobile-data restrictions separate from WiFi:

  
NetGuard: List of apps, settings for a specific app, defaults (source: Google Play; click images for larger variants)

As the screenshots show, you can set the default to "block mobile by default", so you don't have to take care for new apps unless you want to allow them.

Third: I find it very strange that mobile data are summing up if you have disabled them, that shouldn't happen. But using a firewall app, you should at least be able to reduce that massively.