Yes they do. Which means they aren't liked or trusted by Ulthuan as a whole, and using them to score political points is a poor prospect because the people you're trying to appeal to don't like them.
Any evidence for the forces of Ulthuan being 'tied up elsewhere'? Yes, the Shadow Warriors do respond to calls for aid, but explciitly do so in ways which leave them with little credit. That's literally what I said.
The Gates are there to prevent armies crossing the Annulii Mountains, there's nothign saying they're built for raids. Neother the Druchii nor the Asur have infintie troops and if anything I would expect Ulthuan to be more capable of bearing casualties, given that Ulthuan is largely a super-fertile paradise with minimal threats while Naggaroth is a cold wasteland whose people are near universally despised.
And everytime the Druchii actually occupy Nagarythe, the Asur raise their armies, drive them out and once again burn Tor Anlec to the bedrock. The raiders however aren't occupying the place, and it's too costly to drive them out when it's impossible to prevent them from returning. Sure, if the Shadow Warriors weren't contesting it the other Kingdoms would have to, but it's not as big a threat as you seem to be making out.
Those are all still part of Ulthuan (as a political entity, although not part of the island proper). And the last force Ulthuan deployed outside of that which we've seen come up was prior to the Great War, when Finubar sent troops to hunt down Morghur to win points with Estrielle and the Asrai.
Honestly this is kind of getting tiring, this feels like we are arguing in parrallel viewing the same point in different ways.
1. I think this is conflating a couple different strings so I am just going to say this. The Shadow warriors are not political or meant to be. Being liked or trusted has nothing to do with it. They are respected, feared and driven. The ones who raids Naggaroth to free slaves, who make the Druchii fear to step on Ulthuan. The best way I think I can put it is that they are respected as the ones who make the enemy hurt the way the enemy made you hurt. Or perhaps you can view them similar (if not exact) to someone like
Pretty Boy Floyd was he a bad person? undoubtedly he was a murderer and a thief. But to the thousands who's debt he set on fire? You think they care that he shot people? Thousands were out of house and home from the banks actions. It isn't nice to think about but everyone has that little voice that will say "good they should suffer just like they made me". People aren't exactly going around crowing support for them, but they have at the very least, silent acceptance and acclaim by some.
2. Ulthuan forces are noted to be active on the sea's constantly, there is the navy, they have two fortresses constantly besieged by Druchii, then another two that need manned.
3. The gates are there to prevent access, AKA: preventing raids. To do so they need to constantly be manned and repaired, rough guestimate of numbers would be about 1500. Also as far as Ulthuan having minimal threats... The Annuli mountains are full of monsters, and dark elf raiding parties. It is noted that soldiers who man the gates are respected for surviving in general. This seems to indicate a certain level of attrition.
4. The whole point of a guerrilla war is area denial, that is what the Shadow Warriors are in effect. Because the area is consistently contested niether side gets to build it up. No Asur fortresses, no Druchii black Arks, and as far as other Asur driving them out that isn't how it works. The only official military that answers to the PK (that is able to be deployed solely by him) is the Sea Guard and those seconded by the kingdoms for campaigns. Without a figure pushing them to contest the landings they have no reason to contest Nagarythe, the whole reasons the Gates were built was to keep the threat contained. In the princes eyes they have done their duty, no reason to slog in without someone making them.