Pre-KotFT, I spent maybe $15 on HS (the welcome pack after I had already been playing a while, and a few arena runs when I was starting). I ranked up fine. Definitely don't need to spend a cent to get past rank 18. I've still only spent $15 or so, but I bought the KotFT preorder with WoW gold, which is worthy of an asterisk.
I'm sure it is with aggro druid or pirate warrior. I don't know if hunter would work still.
I think Toast did it with hunter fairly recently. It is likely harder than it used to be, but still doable.
I think established players doing it is fairly irrelevant though. They already know the basics, know how to make a deck, and most importantly know what they can dust and what they should keep, which is a big deal when it comes to these budget legend decks people tout. New players don't really know that. Something like PW is relatively cheap compared to its power level, but that doesn't really matter if a new player doesn't understand that they pulled a crap legendary in their first 10 packs.
I've heard some pros and streamers suggest adding a little bit of straight dust to every pack, and I think that's a good idea. HS has gotten better for new players, but I think there's still more they need to do and I'm sure they are workshopping ways to handle that.