Bare with me cause I am going out on a limb and making this up as I go:
Since Quake was mentioned I tried to compare these kind of scenarios to Quake dueling as it simplifies the matter by having only 2 players. Why does Quake not have complete stalemates, or why does Quake have semi-stalematey situations that still remain interesting for both the players and the spectators?
In TF2, having CP3 can be compared to being in control. Having more round wins than the opposing team in TF2 is the same as having more frags than the other player in Quake.
On most maps in Quake it is not uncommon, and possibly even expected, to see a player out of the lead and out of control make a comeback, regain control, and make frags. Why does this happen? Because being in control in Quake is an active process and requires the player in control to give up positional advantage to maintain said control (circling the items means you will be traversing positionally disadvantageous areas). Also, as the player in control, staying completely passive and defensive between the major items allows the player out of control to gradually even the odds (stacks) by circling the minor items. Whichever the case, there is almost always some kind of possibility for the player out of the lead and out of control to make a comeback (up to certain limits of course). To retain game dominance, the player with the upper hand is forced to take certain risks constantly.
Now compare this to TF2. Being "in control" (having CP3 or CP4) is a passive state. Staying in control does not require the players to sacrifice superior positioning; in fact, the opposite is true: Keeping superior positioning is what maintains the control! Positioning in TF2 is a goal in itself (getting the position on CP5 means you win a round point) and not a means to an end, like in Quake. The means to an end in TF2 are the frags, as opposed to Quake.
With the nature of comp TF2 being about scoring through position and frags only being a means to an end, how can we ever fix this problem? Stalemates are indeed a natural state of the game; they are a result of the scoring meta.
Without changing entirely how comp TF2 works, I think the only solution can be found in changing map architecture. Why, when we talk about stalemates in TF2, the maps we see recurring are snakewater, gullywash and granary? All these maps have no open chokes between CP2 and CP3, and CP3 often offers a huge positional advantage (mostly height advantage) towards CP2.
Especially on snake, if the team in control (has CP3) has good spam on the chokes, its impossible for a roamer from the CP2 team to try and make a force play. Its near impossible for the demo from the CP2 team to do good spam (standing at the bottom medpack near green its hard to get a good spamming position without receiving more spam damage than you deal, and standing in saw room you will already get spammed before you even get close enough to the doorway to get a peek onto CP3. Compare this to badlands choke where the team that has CP2 can lob pipes/stickies through choke with enough room to dodge the counter-spam, and can launch bombers to make a play relatively freely.
So obviously badlands does a better job than other maps (of course, it's the best fucking map), but I propose we should experiment with new maps (or modified maps) that enable teams out of control to make plays more easily. This means that transitioning from CP2 -> CP3 should actually provide a positional advantage instead of disadvantage. The team that has CP3 already has logistical advantage (spawns) and another advantage for simply having a buffer before they can lose a point (if you lose CP3 you still have CP1 and CP2 before you actually are put back score-wise).
This means that we could experiment with maps being designed with an overall valley/dip style layout, where CP3 is the lowest point on the map, and where CP2 is elevated over CP3, so the team out of control can push with positional advantage. It should also be relatively easier to spam from CP2 to CP3 than the other way around. With this layout in mind, there is incentive for both teams to push. The CP3 team will have a worse position and will draw the shorter straw spam-wise if they keep that position, and will therefore eventually risk losing control of CP3, so it's in their best interest to push towards CP2/4, also just for the sake of eventually getting close to CP5 and actually scoring a point. The CP2 team will be incentivised to push for the sake of already having positional advantage and wanting to get out of their out-of-control (cp-wise) situation.
I hope that some of this makes sense and maybe some smart mapper will take the effort to experiment with these unorthodox map layout ideas.
note: I don't play Quake, I only watch a lot of streams and vods, so I might make some noobish or short sighted observations about its meta.