You've run into a common problem - "Party RPGs with non-Party Characters". Same Page Tool can't fix groups who want different things, and it also can't fix game design that works against it's own game premise. You have a few options:
Class Limiting
"Hey, we're playing X kind of game and these classes/types in this
game don't fit that. Can we just not use them for this game run?"
Games that usually have classes antithetical to their goals usually also have a pretty broad set of class selection, so it's usually not too bad in terms of choice limiting. The other half to deal with is the social contract of your group.
(There's also a subset of gamers out there who deliberately pick the most contrary ideas to what you state the game is about. "Dude, why do you have a Navy Seal character in our game about civilians running from monsters?" Those players are their own problem...)
Building with Limitation in Mind
"Hey, for this kind of game I want to run, these kinds of characters
will need to fit these kinds of situations. Can you spend your
points/pick your skills/build your powers to better fit this?"
This is a relatively good option - you can get stuff like "combat rogues" and such that are better designed for situations rather than splitting off. This depends a bit on the system's ability to allow customization or choices within the class system, and also lets players know up front what they need to consider with a character class build.
Non-Party Play and Strong Pacing
If you can run a game which isn't dependent upon a party structure, all those character classes generally work fine as long as their goals and concepts line up. In these kinds of games you need to be able to cut scenes relatively quickly, not spend a lot of time on wasted scenes and the players need to have good goals to aim for.
That said, usually systems that are more mechanically light work better for this than ones attempting to balance out a lot of abilities, though games like Burning Wheel or Blade of the Iron Throne can work fine for it, mostly because the basic resolution systems allow for quick play and give good goal-building tools in the form of Flag mechanics.
Playing short form requires several shifts in technique and approach. My group typically does 2-3 hour sessions. A 4 hour session is a marathon for us.
Drop the Filler
The first thing to do is let go of filler material. Filler material includes setting up adventures that are "clue to clue to clue to oh actually interesting development". This is the default for a lot of play, and it wastes time. Let go of having the players play out haggling for everything, or explain each store they go to for supplies, and so on.
If there are encounters that would take up time but not actually provide interesting choices or hard implications? Cut them out.
There's also a lot of time wasting in putting in situations that make it unclear what the players should even be trying to do. "Where should we go next?" "I guess we keep searching until we find a clue of what to do next?" etc. Videogames used to use mazes to add extra hours of gameplay as a filler device, and this is the same kind of thing.
Let players get to the fun interesting stuff without this obfuscation and play not only goes quicker, it's more fun period.
Scene Framing
Scene framing is the next step. You no longer say, "Ok, where do you want to go next?" as the default question. If you know the general thing the players are doing, you skip up to the next part and get right into it.
For example, you know the players are trying to track down a cult:
"After three days of talking with shady people, listening to wacked
out babbling of street preachers, and having to do some small trading
in hallucinagens, you finally find out where they're having meetings.
At dusk, you find yourselves outside an abandoned church where several people died of plague years
ago..."
Think of how movies or tv shows will give you a montage - so you can skip the legwork and go RIGHT to the interesting stuff. This alone cuts out a LOT of time wasting material.
The Rules you use matter
If you're playing a game where combats take 45 minutes... well, one combat will take up a significant chunk of your time. Understand that many of the older games and traditional games expect you to be playing for 6-8 hours as session, so if they expect 5 combats in a session, and you've got 3 hours to play... it's not going to fit.
If you're playing a game where conflicts of all types, take 5 minutes? Well, then things move much quicker. So pay attention to what rules you're using and what their impact is, so you can plan appropriately.
Getting into character, exploring the world, etc.
I've been playing this way for several years now. My group gets into characters quite well, because the focus of our play is on characters, their issues and personalities, which is because a lot of our game time is putting the characters into crisis points and fun interactions.
I find a lot of people used to playing with lots of time, think the solution to character or world development is simply pouring on more time, when in fact, it's about directed and focused play. I've had several people say, "We've done more in this two hour one shot than I've done in entire campaigns, for years."
That's because I cut out the filler, focus on the characters' choices and actions, and try to give situations that are at turning points. Each scene should matter and have some impact.
Best Answer
I find GM screens put important tables front and center for reference. When a GM becomes familiar with the tables, so then does the GM screen fall. Thus GM screens are often more question of GMing style and mastery of the rule set being played.
To some, GM screens are a symbol of authority and must be "torn down."
As smartphones and tablets augment play more and more, I have found secret rolls can be made silently on these devices and not behind a screen.
Play style, and table dynamics, will often dictate the use of a screen, as you have found out.