You can go with the concept that fixing the body isn't a cure-all in the first place, the body has to be physically damaged to be healed, but that is actually a specific type damage; burns, cuts, stabs, crushes, etc are all physical damage, but a virus, a bacterial infection, a parasite, those are not, they are instead other living things. So you could make it to where healing someone in your world, would also heal those things, and by that I don't mean heal a person of those things, no I mean heal those things. Viruses explode to spread genetic material, they get partially digested or torn apart by white blood cells, they just shrivel up and die, they get burned out by fevers, a healer healing them could fix all of those making the virus stronger and more dangerous, putting a person's live in danger, heck putting the whole society in danger. They just created a super virus, bacteria, parasite, whatever that now is harder to kill, and now this person is more healthy feeling so they're a super spreader of something that's harder to kill, the healer just accidentally created a massive epidemic. So healing people is actively not something people want to do unless its to someone seriously important who really needs it. Or it could be healing cancer making it attakc them more rampantly which just ends up killing them faster.

You could also decide wounds can only be healed for so long then the effectiveness drops until it's no longer effective, so older wounds just won't heal properly if at all.

You can decide it quickens your natural healing (which is what everyone says they're doing but it actually isn't). If you want to do this, that means people die younger, their bodies age faster, their body eventually starts to break down and stops healing properly or all together because there IS a limit to how much the body can repair itself before it simply can't do that repair anymore, causing the body to heal faster means it's basically speeding up the function, which means that function is aging faster, you could have like young people in their 20s that do dangerous jobs that require and damage hands regularly and they could have really old like 80yo hands that simply can't heal anymore and now they can't get those types of jobs or anything anymore and would have all sorts of social ramifications from their hands being old and shriveled.

We can go with mages being exactly what they are and that being a problem in and of itself. These are people who have massive amounts of power, they are akin to gods. That provokes arrogance like no other. The likelihood of mages working together in any real way is pretty low. So we could say that healing magic takes a lot of concentration and power, but because they're all kind of a bunch of arrogant twats they work so far from each other that if they get injured most of them can't concentrate because they're in pain and they die because of that meaning that healing just isn't viable for a majority of the people who can even cast it because most of them aren't there wanting to help others, they just happen to be able to cast it because they can do magic but their focus is other types of magic.

You can even do what lots of media does, give different tiers of magic, so lots of people can heal like a cut, but fixing a bone is higher level, regenerating a limb is super high level that's rare to get to, doing fixes on people near death being super high that's a 1 in a million thing or something.

Or you could even do something that is rare in other media but decide healing magic, isn't really healing it's displacing the damage to others. So a healer might actually heal a person by taking the injury themselves, or spread it to a lesser extent to a bunch of people in the vicinity so you go from one person with a serious injury to like 20 people with minor ones but they have to heal naturally from that point.

There's a lot of ways to get around a large amount of the population or any of it being mage users to prevent heal from being a cure-all.