TL:DR flamewaker mage is no different than most aggro decks, has much higher skill cap, more easily counterable and can gas out.
I don't really agree with this take. You can play against aggro in a way that makes sense, and is possible without specific tech. You keep the board clear. You manage your health as efficiently as you can. Those are the things you do against aggro.
But against APM Mage there's no way to do either of those. They don't play minions. Unless you're a Warrior, your health isn't going above 30, which is a totally manageable number for the deck to burn away at turn 5-6. Playing aggressively to take them out before turn 5-6 is the only way to beat them, which isn't how you counter aggro - because that's how you counter a combo deck. But a regular combo deck would need to reach 10 mana (give or take a bit) to combo off, even then they'd need some level of prep.
I don't know what you're saying with this whole "there's no recovery" thing, the deck is like 75% spells that generate more spells, either by drawing them or randomly generating them. It has either 3/4 minions depending on whichever variant of APM Mage you're playing so Refreshing Spring Water nearly always is free or plusses you on mana. Sure, the deck can burnout to like 5 cards after 1 Flamewaker turn and still not kill you (anecdotally I had 2 full boards as even Paladin that Flamewaker Mage managed to burn through and kill me with anyway, so going wide didn't help at all) but if they keep a Flamewaker in hand they can still go again next turn and clean up.
And the "a God Hand is required for the deck to win" argument doesn't really hold up to scrutiny. If the deck was purely highroll, then why is it the most popular deck on Wild Ladder? Why is it doing so well if it only does well when you're lucky? Is every APM Mage just like, the luckiest person in the game? Sure, I've won maybe 2 games against APM Mage, but only because they were really UNlucky. If an absolute terrible brick-hand is the only game state where APM Mage can lose, it's not exactly "God Hand required" - it's more like "literally anything but the worst possible hand required".
1. You keep the board clear is nice in theory until you you play some games and find out that aggro decks usually outdraw control decks by a mile, hence they can produce multiple boards while you have limited and usually expensive removal. Also "No tech required against aggro" well, mate, your board clears are your tech against aggro. Think about it if there was no aggro at all, would you run low cost board clears in your deck and have them sit in your hand in mirror matches?
2. On the no recovery - if you use tech card and pull/kill a combo piece OR APM mage doesn't kill you on in one turn there is no recovery in 99% of cases.
3. On the god hand - god hand to win on turn 4 is what I wrote, not on turn whatever.