Yes, I think the same. In fact, many decorators prefer water-based. Unless the arbour is a pretty solid, substantial one, I don't think I would have used gloss at all.
Easy to say in the land of Hindsight though. I can't foresee any fungal problems, but I would worry about cracking. Smaller, less substantial arbours will suffer from movement. Constant expansion and contraction from heat and rain, plus wind movement.
Next time you might want to go for a finish that's a little more flexible. Modern opaque, coloured stains are much easier to apply... with minimal preparation. After around seven years, they'll just need a wipe with a cloth, then a quick re-coat.
If you've done all your preparation, then you'll be fine. Just keep an eye on possible cracking. Once water gets into a crack, gloss paint will deteriorate quickly.