There are 20 zero cost cards out of 132. Almost all the support cards cost zero, also modifiers, utility cards. Based on the classes:
1 aqua card for energy gain, important mid/backline aqua card
3 beast cards, draw, morale and energy
4 bird cards, attack buff, free damage for damaging yourself, cheap fear card and cheap combo card
6 cheap bug cards, mainly square teeth, laggin and leaf bug are part of the meta.
3 plant cards. bidens is important for poison removal, mint is removing stun from yourself or poison from the front teammate and cattail which is a good combo activator for things like leaf bug.
Classes that lack base health or speed, got more of these cards to make up for it, but they can be used in any body type and allow good combos. 7 of these cards are on the tail. Nimo, luna and cattail (with leaf bug) are used for energy gains. 3 on horns, these are all meta picks, the cuckoo aquas gaining popualrity, the laggin is part of terminator builds, and leaf bug is a popular front or back plant. 3 mouth cards which you don't really see (pincer with the other discard quite good) and 7 back cards (usually defensive part). Hero is one of the worst cards in my opinion. And all the other 0 cost back cards require a really specific build.
You can see from the list that they are not for damage, other than post fight. There are 8 parts for mouths that deal more than 100 damage (without modifiers) and there are plenty for every other body part, but only post fight is 0 cost, so a normal first turn nothing really goes above 400-450 health damage, but post fight would be able to do above 516, which is the maximum health without shields.
There are 5 cards that require 2 or more cards comboed with them, or 3 energy at least, so normally you would need some sort of energy generation and quite a good draw to be able to pull it off. Interestingly enough 4 of them are targeting cards allowing a backdoor and they are all on the back of axies. While I haven't saw many turnip combos, ronin, perch and trispikes are all on the expensive side and they all usually use a 0 cost card, which as you can tell can't be the back part. And as you can probably tell, chomp is part of the terminator build. By using 0 cost cards, you can activate these combos and targeting cards for cheaper.
So why would you use 0 cost cards if they are weak in attack? As you know, you only get 3 energy and 6 cards first turn. This means 3 of your cards are flexible and you can only use half of it. The following turns you get 2 energy and 3 cards.
As you can see, you run out of your whole 24 cards by turn 7 if you still alive, but you still can only spend 15 of them, if the opponent doesn't interfere. What if you only got 2 axies alive? The math is still the same, 2 energy 3 card, only that you got higher chance to get repeating cards the more you spent. Matches are flexible, you might get a different card draw, you might use one card in one matchup but not in the other. You can modify this curve, by:
generating energy (luna, nimo+nimo, leaf bug+cat tail)
carry over (iguana,kotaro) you spend one but you gain back if the condition is met and it depends mostly on you (you can see buffs or speed order)
wager energy (ronin+imp, serious+cattail/leaf bug, carrot, rice, antenna, watering can) - you might steal energy but you spend energy it, so it depends on your opponents too
destroy their energy and yours (goda)
drawing or discarding
0 cost cards.
So even if zero cost cards do less damage, you can get rid of cards faster. You don't want too many, as you will run out of cards and you can't do anything with your energy, usually you got 3 starting and 1 each turn so you would be able to use 1-2 cards on zero cost per turn with no card draw. If you stall a turn to gain more energy then 2 seems optimal. You lose cards if your axie dies, so this means you need to spend it. Usually front axie dies faster so having 0 cost on him is actually good. When he dies, the pool gets empty and you will get 3 cards out of 16, but only 2 out of 16 is zero cost so your damage won't fall and your hand won't get empty. You could rush it to kill one more axie but then you might need to wait a turn. With one axie left you get 8 cards and usually you need 1 energy gain 1 health/heal, 1 damage, 1 utility. So you either gain energy or use a 0 cost to be able to play 1 card each turn. There are multiple options, you could use your back liner to generate more energy and the front to be sturdier using that energy. And it also depends on your opponent, while you can technically prevent steal or destroy, you can't prevent him gaining energy, and sometimes you would need to steal/destroy, even if you don't use it. So there is no perfect distribution, but anything different than 12 of 1 cost cards is good. Probably 2 zero cost would bring situations where you got less damage than you need, so that's why I would recommend 1 per axie maximum. There are also mindgames with ranged/melee attacks, and shields/damage. While some teams use 4 damage cards on at least 1 axie (aqua), usually in game you can predict when an attack is coming so 1 defensive card is good. 1 ranged/1 melee can prevent disabling skills and at least one card have an effect that is good situationally (goldfish is good damage but also helps with speed).
The other use of zero cost cards is that stun or fear effects will waste you a card, so using a 0 cost on the same axie gets you out of stun, using 1-2 0 cost on other axies gets you out of fear. This can be done with defensive cards too that have 0 attack.
Overall I would say, 0 cost cards can be really useful, and you will improve your team using them, especially against front to back teams that try to maximize health in front and damage in the back. It gives options to beat teams you are not familiar with. But it also makes you more prone to card draw. They will be needed in higher MMR, but I don't think you could ever get out of 1400 range if you don't have one at least in your team(nimo has a chance to counter terminators). Most people who can't even get back to 1200 MMR got no energy gains or zero cost cards and a lot of floor axies got too many of them without combo activators or enough damage or redudant targeting cards.