It is hot as hell here in Arizona right now - Zion is probably the same, not sure if that is a factor for you but just thought I should let you know. Some young guy just died of heat stroke at the bottom of the Grand Canyon trying to do a rim to river to rim hike 1 day hike. I’ve done that hike many times but only once in the summer and it wasn’t enjoyable at all - and this summer is one of the hottest in a while.
Having grown up in Arizona and having hiked in the Grand Canyon many times I can assure you it is a nice sight, but if I were to choose between it and the Sierra, Sierra wins hands down. I hiked the PCT through it in 03 solo - it was pretty much a religious experience. It made hiking through the Southern California and the Mojave desert in June and July seem worth it.
My advice with 20 days, DO THE JMT! You won’t regret it! You will probably need a bear canister - I didn’t use one and most PCT hikers I’ve heard of don’t use one through the JMT but do be on the safe side and to avoid hassle from rangers you should probably get one or rent one.
As far as permits go - I don’t know how hard it is to get JMT permits, but if you have trouble here is what I would do: get a PCT thru hiking permit and if any rangers stop you pretend that you are doing a large 500+ mile section of the PCT and simply started from around Whitney. Personally I never got asked for a permit in this section. But I have to agree with the last poster - most JMT hikers go South from Yosemite - if nothing else ending your hike on top of Whitney seems fitting. Also if you are hiking alone you will probably meet up with less people going north if that matters to you.
If you need more convincing see my PCT gallery with images from the Sierra:
www.pbase.com/aaron111/pct
Aaron