Or we could run it like now and hand out haste to the cards that needs it
That wont happen though as described by the haste change, it will be very specific. I am talking about all characters with activated abilities vs just a few specific ones. I'd say the current implementation needless buffs 1 time spells and removal. Think of potential abilities that could be used from the support zone provided that your character/location did not come into play exhausted:
0, Sacrifice this card: deal 2 damage to target character
1: deal 1 damage to target character
Ability may only be used once per game.
3: target character gets -1/-1 until end of turn
Ability may only be used once per turn (this could also just be worded like the below example)
2, Exhaust: give target character +1/+1
4: give target character Flying until end of turn
4: place the first character in opponent's attack zone in their user's support zone
In all of those examples with the current way you would have to wait until they have been in play one turn and would weaken their abilities so much and limit your strategic decisions. If they were able to activate when they came into play you could have more freedom on your turn and use those characters in place (or in conjunction) with 1 times spells that did something similar.
If an ability is too strong to be used the turn it comes into play make the card come into play exhausted or an exhaust ability if it is decided that that would not be usable the turn it comes into play.