You don't need Hover Boots in order to get to that rusty switch - after playing Zelda's Lullaby on the statue's hand, two chests will appear (on the opposite hand and near rusty switch). You can easily use the Longshot on the second chest and activate the switch.
Oh yeah that's right, so the hover boots are completely unnecessary in the Spirit Temple.
Fact is, the hover boots and the eye of truth help a lot in the Gerudo Training Grounds and the Spirit Temple.
Whereas the silver gauntlets and the mirror shield do nothing in the Shadow Temple.
So while it's possible to do the Spirit Temple first, why would you want to do that?
Well the one thing that is undeniably helpful from the Spirit Temple is the heart container. Also doing the whole Gerudo Fortress area and Spirit Temple area before the Shadow Temple allows you to do the Shadow Temple with 19 heart containers, and IMO the Shadow Temple is the hardest temple combat wise (in my 20+ playthroughs I have died twice, both times in the Shadow Temple).
Also I often do the dungeons out of order and do minimalist runs. My favorite thing to do is complete the game in the order Forest, Water, Spirit, Fire, Shadow without ever obtaining the Zora or Goron Tunics, only collecting one bottle, only picking up heart containers but not heart pieces, and not collecting unnecessary items like the Lens of Truth. So doing the Spirit Temple before the Fire Temple (and Shadow Temple) helps greatly in this playthrough since every heart container you have is 8 more seconds you can spend in the hot rooms of the Fire Temple without the Goron Tunic. Also I like to have as much health as possible for the Shadow Temple since doing it without the Lens of Truth makes it even harder.
I know these are very specific and unique reasons, but the point is that there are many conceivable reasons why someone would want to do the Spirit Temple before the Shadow Temple. However, I would agree with you that the Hover Boots are much more helpful in the Spirit Temple than the completely useless Silver Gauntlets and Mirror Shield are in the Shadow Temple. As for the Gerudo Training Grounds, the Hover Boots do help in that one room but aren't really necessary.