I have been GMing realtime text campaigns for years, and also playing when given the opportunity, so here are some things that I've found to work to varying degrees. None of them are panaceas, and no amount of GM technique will overcome an utter mismatch of interests (such as some people not being interested in cerebral investigative campaigns).
Shorter Posts
Whether as player or GM, avoid huge posts that take five+ minutes to type, as they tend to go wrong in one of two ways.
First way: everyone waits for the end of the post, not doing anything at the time. This of course stalls the scene for five minutes, and tempts others to alt-tab to some other activity. Then there's another stutter moment because now everyone needs to read the whole post before being able to react properly.
Second way: others don't wait. In which case perhaps someone else makes a quicker post (character statement, player question about the environment, or even character action in non-structured/non-combat time) that invalidates part or all of the huge post, requiring a rewrite. Or perhaps the other interactions don't do that, but there's still a stutter in the pacing as everyone stops to read the huge post before being able to continue.
Options for Background Interactions
This is a trick for increasing the overall pacing of back-and-forths even while the current spotlight character's track isn't moving any faster. It relies on the setting having communications solutions - technological or magical - that allow the whole party to be in touch and aware of what some others are doing.
For example, the face may be talking in the spotlight, while the others are quipping back and forth in the shadows, providing ideas and maybe even doing minor tasks in the hopes of aiding the face.
Parallel Processing (with Proper Markers)
No, not the computer kind. Often, the case that it doesn't really matter in what order tasks are resolved. If one player takes too long to think, it can be okay to spend that time on processing the outcome of some other PC's action or a player's question.
This can indeed be confusing if not properly marked. So use @name and/or other markers that make it clear to whom you're replying, or at least in which of several separate locations you're describing an outcome.
In my experience, that significantly reduces the amount of awkward 'silences' in the game channel, but doing the former without the latter can be confusing, and doing the latter without the former is often almost entirely pointless.
Link to Visual Content
A picture is worth a thousand words. A crude but clear map can solve a lot of ambiguities about the topography of a scene, making people need fewer clarifications, and more confident in acting in ways that depend heavily on the arrangement of entities in a scene. Character portraits and scenery images can similarly convey information faster than a long description.
Consider Music; Yes, Music
Yes, you are running a primarily text game, with no voice. But if you have the opportunity to use Discord's audio channel for music instead of voice (and the players can listen to it), it can still serve to convey information faster, requiring less typing. This is of course mostly useful for conveying overall tone, atmosphere, or emotions of a scene: telegraphing danger, setting up a calm or conversely spooky mood, or conveying that something a PC said has greatly saddened an NPC (the latter trick once worked so well that a player regretted that the PC can't hear the musical cues).
