This was driven home to me many years ago when I popped a SIM from a Mexican carrier that had an embedded Dominos Pizza app on it. Suddenly the Windows Mobile phone I was testing had a new icon on it.
For example in many African countries, you have M-Pesa [1], which was at least initially entirely based on SAT.
Also, the API is somewhat limited. "Installing applications" here means "downloading code to the SIM card", which arguably has always been the phone provider's property.
It's definitely not possible to install apps on the application processor OS via SIM-OTA. That would be OS-based carrier profiles, which the OS vendor has deliberately implemented.
If so, any entity with a court order, can install anything it wants on your phone.
Alternatively, any entity it wants can use the sim itself to track beyond the norm...
It‘s not "any entity", though – the provider’s keys are needed to do this, and they can already do much of that tracking using other, network-side means.
If the signing keys are compromised, though, bad things can happen: https://www.srlabs.de/bites/rooting-sim-cards