I was thinking more about the Intimidation case. I think applying the Frightened condition is a good way to model fear. But how long should it last?
The closest analog here is Grapple, which is Strength (Athletics) versus Strength (Athletics) or Dexterity (Acrobatics), target's choice, and imposes a condition, Grappled. But Grappled doesn't seem as bad as Frightened, and the grappler has to maintain it by not moving away, and it takes up one of their hands. OTOH Grapple isn't a full action (it can replace just one of your attacks) and the victim must use an action and make another check to break free.
Since Frightened seems more severe than Grappled, the easiest solution is to say it lasts for 1 round (until the end of the intimidator's next turn). This gives the intimidator 1 turn to take advantage of the condition. Then the victim "breaks free" automatically, no action and no check. If the Intimidator wants to keep it going, they need to keep using their action to make Intimidation checks (which the victim will have disadvantage on if they can still see the intimidator).
Out of combat, that seems a little weak. One easy solution is give the victim a temporary flaw of "Intimidated;" if the PC acts out that flaw, they get inspiration. If the victim is an NPC, the DM just role-plays it.
A solution that works both in and out of combat is: the Frightened condition lasts until the target makes an ability check or attack roll, or takes a short or long rest. So if an NPC intimidates a PC during a peaceful negotiation, the PC becomes Frightened and is going to take disadvantage on their next check or attack if the NPC is still in sight. If they wait until the NPC is gone, their next roll won't have disadvantage, so they'll get out of the Frightened condition for "free." This works in combat too: an NPC intimidates a PC, and then the PC's options are to go where they can't see the NPC and make a check or attack roll, or else just suffer the disadvantage on their next one. If the NPC dies I'd rule that counts as being out of sight so the PC doesn't suffer the disadvantage then, either.
Either way, the using Frightened condition gives the PC options for how to handle that, instead of having the DM take control of the PC or forcing them to act a certain way.