I am a Linux user with a frequent need to ssh into remote computers and keep my session there running while disconnecting. And then later connect back to it from the same or a different 'client' computers. There are a number of programs, *terminal multiplexers*, that solves this problem. I frequently use gnu screen, tmux, etc.

However, they all seem designed around the idea of taking over rendering of the text surface available to them to do a form of 'screen management'. In doing so they override tasks normally handled by the terminal window on the client computer (i.e., the xterm, gnome terminal, etc. from which I am running the ssh). This means that once I start the multiplexer, I can no longer use, e.g., the scrollbar to access the scrollback history, or the search menu of my graphical terminal. I can no longer mouse mark and drag in the window and have it scroll back by itself, etc. Instead, these terminal multiplexer provide their own keyboard-based user interfaces to handle such tasks. I would much prefer if I could keep using the UI of my client terminal as usual.

Hence, is there a terminal multiplexer that provides the function of a virtual tty, which one can attach and detach from at will, but when one is connected it simply just outputs all text into the console "the normal way"? I.e., without doing any form of "screen management"? (I'm imagining that it could still keep a log of recent history internally, so that when one connects, it can spit out this log to provide context to a newly connected client.)

(Answers that point to ways to configure well-known multiplexers to do this are of course acceptable.)
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I think X2Go fits your requirements, based on NX3. <https://wiki.x2go.org/doku.php/start>

I use it and while it's not perfect, it's the best I've found so far. 

It may be in your distro's repository. X support only, no Wayland.

Uses ssh, logs in to existing sessions or creates new sessions using remote system accounts, and has a cool "seamless" application mode for running a remote program in a local window. 

Problems I've encountered are mostly with keymaps. i.e. in session mode, some keys don't register correctly for me.
You might want to clarify what Kind oft "remote desktop" you want:

1. Mirror the screen, i.e. interact with the user session which is currently active on the physical screen.
2. Open a New session exclusively for the remote user.

1.) is the target oft VNC, but also TeamViewer and the likes. It is useful for remote assistance, but not well suited for remote working. 2.) is RDP, and NX as well, IIRC. This is useful for (multiple) remote workers.

Back in the days, remote working in UNIX was very common, with thin Clients connected to a central Workstation. After all, that's what the X window system client server model was made for. This is now very uncommon. If you're looking for some remote assistance tool (case 1), then I think VNC+SSH is still the best option. If you're looking for something like windows remote desktop services, then I am afraid there is none.