The phone must run the apps because doing so on the watch would (1) consume limited RAM, and (2) require the watch processor to do the thinking which would seriously compromise battery life. Apple wants the phone to do the heavy lifting and "stream" display changes to the watch. So the watch is basically a "dumb terminal" for the apps.
Supposedly, Apple is working on on-board app capability but looks like they have optimizing to do.
As for lag,
this website discusses the nuts and bolts behind what's going on pretty well. Everything is streamed from phone to watch over Bluetooth Low Energy (unless you are at home on wifi) which has speed limitations. Developers are still learning how to optimize around that, which is discussed in the linked site. Thinks like avoiding too many jpgs (looking at you, Flipboard), pre-caching as much as you can, and relying on vectors more than bitmaps. It's promising to read that they are figuring out and it's just a matter of coming to terms with the differences.
When I get my watch, I plan to install some apps but only the most important ones for glancing. I'm not going to go overboard. I still have a phone in my pocket for the more involved interactions.