"Is it likely to re-sprout if we don’t get every last bit? Any roots that go well under the deck we cannot get at unless they just pull free. Or will it die if we just saw off the trunk? That would sure be easy."

I would plan to paint the sawed-off trunk with an herbicide for brush in concentrated form. Wisteria does tend to resprout, including root suckers, so keep an eye out even if you herbicide it to be sure it was really killed. You might need to use screws or something equally reversible on that spot on the steps so that you can deal with sprouts if needed.

I grow clematis, honeysuckle, and climbing hydrangea. I wouldn't consider climbing hydrangea the best choice here since it is a really large, really heavy vine, and mine took something like 7 years to bloom. The support needs to be substantial; mine grows on a concrete wall. The bloom period is also relatively short compared to the other two. It's a great plant, but to my mind more for a short season of high interest and background the rest of the year.

My clematis bloom for 3-8 weeks each, depending on the variety, so if you use a mix of kinds you can have a longer bloom period. Just hard prune them all.

My native honeysuckle, Lonicera sempervirens 'Major Wheeler', blooms profusely in late spring, but continues to send out blooms on new growth until hard freeze in the fall, so a really long bloomer. The hummingbirds like it and the foliage looks healthy all season. There are some other attractive honeysuckles; be sure to avoid the invasive ones (particularly L. japonica) and choose one with good disease resistance for attractive foliage. Here is Major Wheeler in May. It is a good-sized vine, so don't count on it staying small like this.

Wisteria has been chopped! The roots proved very stubborn and we got nowhere with them. “Pulling them out” was a joke! Nothing budged. Considering they are essentially growing in / through a box made of wood, we couldn’t hack any further without damaging the framing of the deck.

I will have to go the herbicide route. Is Stump Killer what I am looking for?

That honeysuckle is SO attractive! That's jumped to the top of my list! I didn't know they came in red. I've only seen yellow with pink. You have it growing on an obelisk? The pergola is more than big enough for a large vine. The wisteria was certainly big enough.

My favorite nursery lists quite a few kinds. I will start a
new thread to ask about those.

Stump killer or brush killer. Read directions and follow for stumps, and wear recommended safety gear - goggles and rubber gloves at least. You want to paint it on the fresh-cut surface, so you may need to recut since the xylem or phloem (can't remember which one goes down) gets plugged relatively soon after cutting. (sort of like scabbing over on a cut.)

Thanks for the added details!