I'd be cautious of Beeper. I guess my biggest concern personally is that it introduces an unnecessary middle man, and its closed source and proprietary, so there's no way at all to verify any claims they make.

To be charitable, I do think their claim on re-encryption of messages through the services themselves is true, but I think the bigger concern here is that whatever messages you send/receive also go through Beeper and not only services like Signal/iMessage/WhatsApp, and I'm unsure if Beeper encrypts the data they receive, or what access they have to it/how they handle it. That's my biggest issue, as it would effectively make end to end encryption on these services useless if Beeper just had full access and could see all of your messages and any metadata, which is what I suspect is the case.

What is the "pain" you're experiencing? You having a hard time with clicking the notification every app sends you? Difficulty identifying their unique app icons? Must be tough.

By definition if content is re-encrypted then at some point it was unencrypted and available in plain text.

I can't speak on anything with beeper but a similar service that is self hostable is matrix-synapse. I mention self hosting as an answer to your concerns of safety but the other side of the coin is this isn't introductory stuff, if you don't have a background or general enterpris experience with this kind of thing i don't recommend it. If you do then this seems like a good fit as you can run your own bridges into the matrix service you're hosting so all your data stays on your own systems.