I've been in SF for 3 years now and haven't really heard anything that impressive about Kleiner, but looking at Crunchbase, they've led the A for Rippling, the C for Plaid, the D round for Intercom, the B round for Figma, the E round for Peleton. They have a bunch of other impressive logos that they've also managed to lock in.
To your point about the web - they clearly still have it. Yes, they're out of the loop and aren't getting the early stage deal flow that they used to (to the benefit of many others), but they clearly have a deep moat still and can get an ok return from that.