You don't need to attach trellis at all if you plant self-clinging climbers - they will creep up the fence themselves. Self-clinging climbers just do their own thing - eg ivy, which is also evergreen - & there are some beautiful variegated varieties. If it's quite shady, another self-clinging one is climbing hydrangea. Parthenocissus too - though it is more vigorous so might take over a bit. Both ivy & climbing hydrangea are easy to trim to keep in check once they reach the top, so would be more manageable.
You'll have more choice with twining climbers though. Still no need to attach trellis to a perfectly sturdy fence though - it's not necessary. Just attach horizontal wires with vine eyes (look online for details of spacing, wire thickness etc). That will allow you to grow clematis, honeysuckle, wisteria, jasmine or trachelospermum - though it depends on how much sun your fence gets.
If you don't like the appearance of the fence, I'd go for predominantly evergreen climbers - ivy, Clematis armandii, trachelospermum, one of the evergreen honeysuckles etc. Once they're established you can add large-flowered clematis to grow through them (they're smaller climbers that will only grow to around fence height).
Other options are espalier fruit trees, horizontally trained pyracantha/chaeonomeles etc if you like a more graphic look.
If you want something fast, buy larger specimens. Don't crowd them too much though as they will grow faster than you think!
If you want something visually striking, you could always try some of these ideas ...!! The trained pyracantha is faster to achieve than you might think - & he's done it against a fence just like yours - https://herbidacious.calamus.graphics/2020/05/29/pyracantha-espalier/
Good luck!