I think the solution to these stalemates is not to offer a different gamemode--the cp_ format offers some very compelling gameplay, from backcaps to uber pushes that don't require killing anyone--but instead, altering the cp_ format. There are several options to help complete this goal adequately. I will assume that the central problem is getting teams to aggress; usually, a defending team has a much easier time. I'll use Granary as my example, since this is generally considered the most stalematey map. I'm also going to assume these changes only apply to the cp_ format; koth_ is perfect and requires no modification.
1. Capping a point no longer means standing on some small piece of easily spammable ground. On mid, this would mean the entirety of the mid is now cappable, meaning that throughout the fight, playing on the middle point would mean capping an objective. The entirety of 2nd, from the upper balcony that connects to lunchbox, to area between all 3 shutters, would count towards capping, as well as a much larger area on last. It also might be worth considering removing the "blocking a cap" feature, and extending the amount of time required for each cap. Defending a point would mean actually defending it, spamming down whoever decides to stand on your ground. It could also mean retreating more quickly if blocking caps is no longer an option; once any slight advantage is gained, from cap time being nearly completed to a small player advantage, the defending team would be forced to leave.
2. Capping a point grants a power up to the attacking team. The best version of this that I can think of is that a team that has just capped a point is granted an overheal bonus. Any team that just capped 2nd would be granted extra heals to push into last. This powerup should be limited either by a timer, a passive heal bonus (every player standing on the cap gains +50 health, which is gone as soon as any normal overheal would drain), or also possibly negated by the defending team touching the point. With any of these scenarios, the attacking team would be encouraged to push sooner rather than later, creating more fast paced games with less incentive to create stalemates.