Basic Shaman is not actually that bad, you have powerful cards like Hex, Fire Elemental and Rockbiter available from the get-go. But the lack of class identity was a problem until WotOG came in and gave Shaman a card that made the Hero Power worth a damn (Thing From Below).
Sort of. Thing From Below is mostly good because of Tuskarr Totemic and Totem Golem. If those cards didn't exist/weren't played, Thing From Below wouldn't be playable (edit: well, maybe it would be. but not in the aggro deck we're talking about). TFB rarely gets much mana discount from the hero power itself. (Which, incidentally, is why you can play Finley and TFB in the same deck; because you have Tuskarr Totemics and Totem Golems and Flametongue Totems).
You can technically cut any card from a non-combo deck but Finley decides a lot of games single-handedly by pulling the Hunter or Warlock hero power. It's effect is comparable to Justicar Trueheart except it costs 1 mana.
Not "a lot of games" and you're certainly not guaranteed to win just because you got Hunter or Warlock hero power. The question isn't "what does it do when I draw it" but "what % of games will this win me as opposed to card X" (like another Flame Juggler, or maybe a Primal Fusion).
And saying it's "comparable to Justicar Trueheart except it costs 1 mana" is like... really? What does that even mean? It's like saying a Boar is comparable to Leeroy except it costs 1 mana. Seems similar, right? Two charge minions, except one's a smaller body that costs 1 mana.
Except again, the difference is that you wouldn't even use them in remotely the same decks. You use Finley in a deck where your hero power is not part of your strategy (like aggro shaman). You use Justicar in a deck where your hero power is integral to your strategy (Control and Fatigue Warriors, obviously). As a result you can cut Finley from aggro. But you probably don't want to cut Justicar from control warrior.