I made two scripts, one in .Net with a GUI for non-devs to grep a server hostname or tag:name in AWS that resolves to an instance ID for SSH or RDP. And another python script doing the same but without the GUI for the dev team. Works a treat.
But you've already explained why it's a little tedious and now I've documented and understood why. Tailscale MagicDNS does all this nonsense for you. Yeah ok thanks for rubber ducking me I see your point now. :)
Additionally, the instance roles are already pre-configured.
There's almost zero overhead in ensuring SSM gets installed on new instances.
One small benefit over TailScale here, I would think, is that I don't have to rely on another tool to gain shell access. Probably a minor win, if you're running a TailScale deployment. In either case, I'd probably want to go with a single tool just to minimize the attack surface area.