Something which removes abilities from a Creature will remove the Haste, but the card will still get Exiled the next time it leaves
(The same older printings of this card also fail to reference the "battle" card type, which didn't exist at the time, but Tarmogoyf's ability still counts that card type.) 2021-03-19 The ability that defines Tarmogoyf's power and toughness works in all zones, not just the battlefield
The latter is really where the value lies in this box
Modern Horizons 3 Historic and Brawl Pre-Bans Historic Harbinger of the Seas Winter Moon Endurance Fury Grief Subtlety Solitude Flare of Fortitude Flare of Denial Flare of Malice Flare of Duplication Flare of Cultivation Enemy fetch lands Modern Horizons 3 introduces the largest change to Historic to date when it comes to card power
(Look at the top card of your library
Anger spells force opposing creatures to attack or buff the power of your own attackers, while cowardice spells make creatures unable to block your assault