I think we should be able to change what stat shards we have equipped from the champ select screen, not the rune page.
Runes are fairly static and specific to the function of your champion, and generally don't need to be changed except in extreme circumstances. However, stat shards are relatively generalized in their function, and are highly matchup dependent, especially for midlaners (who can lane against both AP and AD enemies). Being able to swap from MR to armor or double up on defensive runes should be a fast and simple process, but it's not.
If you have to pick last and the champion you had lined up just got picked by the enemy and you have to scramble to pick a new champion and create their rune pages from scratch (because we don't get enough rune pages to have one up and ready for every champion we play), you don't have a lot of time to open the rune page, make your changes, and save it before the timer runs out. (I also think that if you have the rune page open and unsaved, it should save your changes before dumping you into the game, but that's a whole different rant).
I'm not saying it's impossible to change your stat shards in the time provided, but it would be a nice QoL change to be able to change them from the champ select screen because of how often it comes up and how matchup-dependent they can be.