1. Monument valley
I would skip Monument Valley unless you will already be driving that road anyway. You can't do anything there anyway without paying a big fee.
Places like the Escalante or Fisher Towers are way better than Monument Valley.
2. Arches NP
June is hot, but the short hikes aren't so bad. Add the nearby Mill Creek, Professor Creek and maybe Negro Bill Canyon. Mill Creek and Professor Creek have nice waterfalls to cool off in. Mill Creek and Negro Bill have swimming.
June should be nice for the Subway. Don't miss it, but it requires permits.
4. Havasupai canyon
Very hot in June and swimming may or may not be a good idea. "Partially treated" sweage is dumped into the creek. Personally I would suggest a spring or fall visit.
5. Vermillion cliffs (the wave)
Permit requirements require planning in advance. Get one now.
6. Bryce canyon
June is pretty nice there.
7. Grand canyon
The North Rim is nice in June, but hiking to the bottom is extremely hot in June.
8. Timpanogos / hiking in the Wasatch range
Make sure to bring an ice axe in June. There will be lots of snow.
9. I'm sure I forgot a couple of things .... ?
Escalante area. It's one of the most scenic places in the world and better than most of the places you list above. Water makes many areas really nice in June. See photos from SP below:
This area is way better than places like Monument Valley.
Yellow Rock (and lower Hackberry Canyon) make a nice alternative to the wave if you can't get a permit.