Is it more profitable to let the new goblin types live longer?

Overall I would say no, it is not worth it to keep them alive. What they drop while running away compared when they die is so small that it isn't worth the trouble. Especially since when they run, they could run into a larger pack of monsters or worse, elite packs, that could keep you from getting to the goblin in time to kill it.

I'm not sure how or if difficulty affects the Blood Thief and how many Blood Shards it drops. On Torment 6 I usually get around 200-250, so the single shards it drops while running are so comparatively small.

The only exception I would add is the Gem Hoarder, but only if you're really hurting for gems and can reliably kill it just before it leaves through it's portal.

It might be a viable technique, but for me personally I think it's pointless to let the Odious Collector (the crafting materials goblin) or the Gem Hoarder (the gems goblin) live for long, as they drop items that are quite common from regular mobs too.

I would however try to delay the death of the Blood Thief (the blood shards goblin), as blood shards aren't regularly dropped from monsters and you have to complete a bounty or rift to get a decent amount of them, so having the goblin drop even small amounts of blood shards before it dies can still contribute to farming.