Trying to settle on some offlane carries to learn. Looking for advice on a few for when they’re good

I want to pick three Pos 3s to learn, just so I have enough to handle bans and can pick somewhat intelligently based on what everyone else is picking. I only play all pick, so I pretty much always lock in third, so I can only counter/synergize pick around supports. I'd still like to be able to do that though. I want to make sure my 3 choices don't overlap with each other so I have flexibility. Here are the ones i'm considering:

Axe

Centaur

Underlord

Night Stalker

Dark Seer

Pudge

Death Prophet

• I think I should pick either Axe or Centaur but not both.
• I had a hard time feeling like I was providing much value with Underlord in later part of game. Seems good because of auras but felt like I had to rely too heavily on the rest of my team.
• Night stalker seems decent.
• Dark Seer looks fun though I haven't actually played him yet.
• Pudge is just fun. I know people hate him but he's just so fun to play, even when losing.
• Death Prophet is fun but it seems like she usually isn't a pos3 so i'm afraid to commit to her in case she stops being a pos3 again in next patch.

So the advice i'm looking for is what's a good combination of 3 that's pretty well rounded out and when should I choose one over the other? Like if my 2 supports have no catch, I should choose X. If both my supports have good catch, I should choose Y. If enemy supports have no catch, choose Z etc. Just general advice.