Shimano 6700 Ultegra (and 5700 105) mechs are designed to work with cassettes as follows…
SS Short Cage up to 30T
GS Medium Cage up to 32T
That is for a dual ring setup with a maximum gap of 16T between front rings (ie. a 34/50 compact setup). In a triple ring setup, you have to run a medium cage GS mech, and you are limited to a 30T cog on the rear (but then you’ll have a 30T chainring up front anyway).
Climbing on a road bike is, sadly, more of a case of grinning and bearing it! You can climb most climbs on bog standard gearing if you pedal hard enough, but of course sometimes it’s nice just to be able to relax even just a little bit after many miles in the saddle.
FWIW I swapped a 12-28 Tiagra cassette out for a 12-30 (with a compact 34/50 up front) on my road bike when I bought it, thinking I’d really appreciate the lower gear as there are a lot of steep climbs round here. In practice, I never really used the 30T cog on the back unless I was already exhausted. Swapped it for an 11-28 105 cassette after a few hundred miles, which I found has better ratios overall for my riding, and still just about a low enough gear to get up the short 25%er’s as long as I get right out the saddle.