I would do another 2 apps of glyphosate before worrying about if you've killed everything or not. From what I've seen triv doesn't always go dormant if irrigated in the summer. If the triv is not dormant you should still be able to kill most of it even though it's not spring.

Digging does work well though and If it's a large area and you are still concerned and decide to dig it out make sure you go well beyond the area of triv so you don't leave any stolens.

 

When I've sprayed Poa trivialis with roundup, it dies more slowly then the more desirable grasses, but it does die. The slow kill is actually a good sign that it really is dying. It wouldn't hurt to spray it again, though, weekly, including the day you sow new seed.

Whenever I dig out Poa trivialis, which I have done plenty of times, I always spray it a week before digging it out.

 

Thank you for asking this, I am faced with a similar situation.
In my case it is a high traffic area so I don't want to.kill months ahead, and is deep shade and has gotten lots of rain.

 

When I've sprayed Poa trivialis with roundup, it dies more slowly then the more desirable grasses, but it does die. The slow kill is actually a good sign that it really is dying. It wouldn't hurt to spray it again, though, weekly, including the day you sow new seed.

Whenever I dig out Poa trivialis, which I have done plenty of times, I always spray it a week before digging it out.

I just wanted to clarify, in case somebody misunderstands the above out of context, that it is ESSENTIAL that the Poa trivialis being sprayed with glyphosate is actually active and growing (not dormant) when it is sprayed.

My comment was addressed to the OP's mention that the Poa trivialis seems to be a real slow kill after getting sprayed with glyphosate:

... I'm doing a reno so I hit it with roundup along with the rest of my yard. 1 week later, while other areas of my lawn are good and dead, the Triv looks... mildly irritated.

If it is alive and growing when sprayed with glyphosate, Poa trivialis will die, but slowly.

As an example of this, check out the below picture of a sprayed Poa trivialis patch, where we sprayed the triv and also about an 8" wider area around the patch, in a northern mix lawn. The picture is 1 week after spraying. You'll note the "good grass" is all dead, but the Poa trivialis hasn't succumbed yet. It will eventually, although I like to hit it again at this point, just because I can.

Should I even bother trying to get a chemical kill, or just take no chances by digging it out and replace with topsoil?

My preference is to do both - spray and physically remove. When we renovated our front lawn as a "War on Triv", I sprayed the Poa trivialis starting in September, for a renovation the following August! Below is a view of the patchwork.

 

If it is alive and growing when sprayed with glyphosate, Poa trivialis will die, but slowly.

As an example of this, check out the below picture of a sprayed Poa trivialis patch, where we sprayed the triv and also about an 8" wider area around the patch, in a northern mix lawn. The picture is 1 week after spraying. You'll note the "good grass" is all dead, but the Poa trivialis hasn't succumbed yet. It will eventually, although I like to hit it again at this point, just because I can.

That's an encouraging example. My triv looks "dead-er" than that after a week. I have 2 more blanket roundup sprays in the schedule - I might hit this area separately though in addition just to really nail the coffin shut.