Here's my 2c worth.
Should have moderate bar pressure. Light bar pressure sucks if you are dealing with big surf and taking a few hidings every now & again.
Kite must not stall through the turns at all. So many different "wave riding" conditions (on shore, cross shore, off shore, wave angles etc) that the kite needs to be versatile enough to handle slow turns in long drawn out bottom turns as well as super fast turns in snappy vertical slashes.
Don't lose all the power in the kite, as although most "strapless" riders seem to be bleating about less power, they can simply chose a smaller kite size. I ride both strapped and stapless depending on the conditions and a powerless kite would never make it into my quiver.
Must have outstanding upwind ability to get back into the lineup.
Don't ignore jumping ability as a criteria. Yes, you have other kites in your range that are more focussed on jumping, but I would hate a surf kite that didn't allow me to huck some decent air on the way back out.
Of course you have to compromise somewhere, but don't just wrte off this criteria. I love putting the TT riders at our break to shame every now'n again with some huge boosts.
Stability and downwind drift must be outstanding.
Getting the power delivery through the turns is one of the most critical factors IMHO. You don't want the power to suddenly kick in like a twostroke powerband, but rather to be consistent throughout the turn radius.
Not too hard, but still enough to pull through the bottom turns when riding strapped. Also, the bar feel/pressure must not fade or get vague through the turn. I want to be able to feel what the kite is doing at all times.
Hope that helps?