Best uses for Starpiece

Best uses are during stardust events or when you have 8-9 10 km eggs hatching at the same time.

Last 2 steps of Meltan quest is a good one too.

I like to save as much second evolution encounters from field research tasks as possible, then timing egg hatches and incense while going by bus. If you find a calm moment (this is quite hard), then open gifts. Maybe now you can also do a quick Candela training in Master League and blow her away quickly with Mewtwo for cheap 750 dust.

I end up exhausted after every use of a starpiece...

i use them to double my stardust.

I don't put any thought into them at all. As long as I'll be walking/catching for 30 minutes I'll use it.

The star piece isn't something to overthink.

If you could combine a couple of the things you mentioned it would be great. Obviously stardust events come and go, so if you're low on dust and want a boost outside of one, planning out some egg hatches while you're in a high spawn/gym heavy area that could have raids would be best.

I'd save them for boosted dust events. You get more from that than anything else, even with raids, they take longer and catch fewer pokemon than on boosted dust events.

Optimally, you stack research encounters, save as many evolved forms as you can, wait for a 3x dust community day with WB, and combine those catches with CD catches. You end up netting 2.4K+ dust off something like Blastoise. If you have a Go+ or Gotcha, you can focus on your stack and let the device suck in the wild spawns.

Combine with hatching eggs and BAM.

Events and full egg batches are by far the best uses. Raids and PvP are both extremely inefficient for dust gain.

Since they're hard to get in large numbers, I only use mine during bonus events in situations where I know I'll catch a lot of pokemon and combine egg hatches with them if I can. Community days with the dust bonus are very good.
The only one I've used recently was on the 26th before the dust bonus ended, I went on a walk to hatch all my eggs and caught about 100 pokemon in total from incense and research queue. Solid 60k gain.

When you can catch a lot of Pokémon. With quick catch it’s easy to catch 100 per 30 mins, with or without evolved forms. That’s usually around 12500 dust. Compare to 9 7km eggs at around 13500 dust and 9incubators. Much easier to control and time catching 100 Pokémon in 30 mins. With quest stacks you can hit 110-120 in 30min.