Wider deep rims more stable in crosswinds it seems, wider tires better aerodynamically with wider rims, and wider tires can run lower pressures for better ride and rolling resistance on imperfect roads.

I think that's my understanding.

I started racing in the 80's and it was all 20mm tires and (for TTs) 140-160psi. Today the best wheels are designed around wider (23mm+) tires. This allows lower psi for less rolling resistance and more comfort. You also get better cornering. Get some wheels with wide (25mm) rims (I just picked up some HED Jet6+ for <$900) and MODERATELY wide tires. 23mm Conti GP4000s II will work well; if you want go 25mm in the rear since the rolling resistance benefits are greater and the aero penalty smaller. Specialized turbo cottons in 24mm front and 26mm rear also make a good choice.

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I switched from 19mm wide rims to 25mm rims (Hed Belgium Plus) recently and love it. It rides so much smoother and handles better especially when cornering. I had 25mm GP4000s on my previous wheels and put them on my Hed's. Because the rim is so wide I will probably try 23mm GP4000s next time when replacing them, just to see what works best.

I'm more concerned with width than depth for sure. I wouldn't buy a rim unless it was out side to out side 25-27mm. Tires 25 to 28. One of the most noticeable improvements in performance (all else equal) IMO are larger tires especially when mounted to wider rims v. older narrow V rims. Very unscientific runs but, consistently faster on wider/wider v. narrow/narrow.