Coupla things here:
1. The purchase of the upgrades and the upgrade request are separate. So, buying them is neither necessary nor sufficient for the request.
2. One can only request upgrade for self plus one other person, not two.
3. One must be elite to use the upgrades, so feel free to ignore #1 & #2 in your case.
Cheers.
I'd add a #4 as I was confused with "I would consider paying $360 to upgrade 3 people across 4 segments a pretty great deal." Each $30 upgrade is good for 500 miles and not the entire segment (unless the segment is less than 500 miles). So each individual will need four for just the single DEN-MIA leg. GCM gives the distance of 1,709 miles between the two airports, hence the need for four.
If you have status, you can request to upgrade whichever segment (or all of them) you want.