Clematis would work (type 3 only involves pruning down to a couple of sets of buds in the winter.)

Lonicera sempervirens would be fine on the structure.

Both would benefit from having the grass immediately around the stems removed, a buried edging added to keep the grass away, and mulch added to keep moisture even.

You could also take the corner pieces and detach them to use as trellises, either individually or in pairs to make a tall skinny box if you don't want the actual gazebo. Then you wouldn't have to remove grass; you could just plant them in the beds along your fence or else where in the yard.

How about Boston Ivy?

wisteria

I disagree with the wisteria recommendation.

Chinese or Japanese wisteria will swamp that gazebo and likely bend the metal unless kept in check with extensive and frequent pruning. It also tends to travel to other spaces both via root suckers and via seeding. It isn’t a low maintenance plant!

https://www.gardenweb.com/discussions/4589466/advice-on-my-chinese-wisteria-border-plan#n=37

And even the American varieties can be large and require more attention than the “little maintenance”requested by durkadurka.

in Ontario, wisteria bloom may not be reliable unless in a sheltered spot.

Durkadurka, did you end up planting vines on your gazebo?

Nice! I like the idea of using pieces of the gazebo for trellis material rather than keeping the entire structure.