I've been running Teleport for a while now and it's been great. It can even manage access to things like Kubernetes clusters which is fantastic in my use case. I've been using their free community edition and no complaints so far.
Xpipe https://xpipe.io/ is an alternative it runs and stores your data locally on your machine and not web based. I've been playing with that a bit, it does auto discover Containerized apps and you can sort of exec into them to run commands and also browse the directories of your containerized apps with a simple click in a File type GUI. It uses your OS's default Terminal application so it won't bring any extra with you so it's more native to your OS.
I've been a Konsole user on KDE for a few years now and it's pretty much what I've been used to. Trying out Xpipe now and Termius about a year ago, I can say that Xpipe is stronger in it's ability to interface with my containerized apps (Docker), but lacks the polish that Termius has visually. They both get the job done, but at the end of the day, I still reflexively just hit my Ctrl+Alt+T key combo to log into my machines.
Then, for a whole different take, SSWifty! https://github.com/nirui/sshwifty - Instead of launching an app, deploy this on your server, and then use your browser's session to securely access your sites.
Tailscale just solves so many these types of problems.
With a virtual network, you mo longer need tools that work over the internet - just use the same tools as you would on a LAN.
I've used Hamachi this way on windows since about 2006. I've waited for an Androidi/iOS client, but it never appeared. Glad to see Wireguard/Tailscale step in to fill that gap, and it's self-hostable!
Not self hosted, but Tabby is the closest I've found. But I still don't like it as much as Termius. And from what other, more experienced people have said, Tabby is bloated, requiring way more system resources than a terminator emulator app should.
Also, I asked a related question here if you want to read some other suggestions.
Tabby and xpipe is what I moved to after I was locked out of my Termius account. Tabby is what I use mostly now, there is a server you can run that lets you sync your profiles.