ProtonMail, the iOS app, is now open source, but ProtonMail, the service, is still proprietary and (mostly) paid.

It's somewhat comparable to using Thunderbird (open source) to connect to GMail (proprietary). GMail might impose some restrictions regardless of the open source status of the client.

There are lots of open source clients for commercial services. You’re not entitled to connect to other people’s hardware for free, even if the software is open source.

Open source (or free/libre software, which is the more appropriate term in this context) guarantees you the right to modify the software to your liking, including the removal of that signature. It does not guarantee fitness for your purpose or usability out of the box.