Oh so doing it without the set equipped works? unlike the rest of the upgrades?

I didn't science it myself (I just left the purple gear equipped while I was working on other characters, and when I switched back later the masterwork was unlocked). But I have heard that you don't even have to have them equipped - once you do the activity (e.g. killing 10 powerful enemies) on another character, it unlocks the masterwork for the character that already has the purple.

Once you have the purple set, you don't need to masterwork them before Solstice is done right? I can save them and masterwork them 4 months from now? I know I should do 1 for the Triumph, but other than that, go onto the next set right?

According to what Bungie has said, that is correct. Note that you'll likely masterwork some of your first set while working on your second and third (10 powerful enemies, season 7 valor reset, and strikes with clan members). You might want to wait on the 2-person shattered throne and 200k nightfall until you have all three sets, so you only have to do those once.

Otherwise you would be screwed, because Shattered Throne won’t be back until after Solstice is over.