Why doesn't support take heal and adc exhaust?
the adc will be doing the fighting, if he or she needs to duel someone the damage reduction should help, And the support is there to.. well, support. so it seems like they should carry the heal
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Two reasons.
First, the support can use exhaust for ganking. Exhaust isn't helping much if it's just sitting on an ADC farming alone, and it's not as likely to save their life as heal is.
Second, having heal on the ADC guarantees that they'll get the heal in a teamfight. If you have it on a support, you're guaranteeing that the support gets healed, oftentimes along the tanky frontliner who absorbed some initial burst. That's significantly less valuable than guaranteeing health to a squishy.
There isn't usually a great reason to give exhaust to the ADC, or to give heal to the support. Neither class can make ideal use of that.
In my experience, the movement speed on heal tends to be much more useful on the adc, especially in soloQ. And given the range of exhaust, I think it's preferred on support because they usually have an easier time applying it as needed - it's also better for roams, and receiving ganks as support.
On a side note, I think as a support player, you typically have a better eye for when that exhaust needs to go down and when you can spare it because of your peeling/chasing cooldowns. ADC tends to be prone to CC as well, so having a support being able to exhaust assassins is helpful.
because even though 
is a required spell for almost every champ, riot refuses to make the necessary changes.
And the support is there to.. well, support
I think you misunderstand, exhaust has high utility. Heal is more defensive, and thus should be on the carry in case of emergencies for the MS boost and the heal (your support isn't going to be there all the time, mind you!).
Secondly, it's also due to range. Supports tend to take exhaust and occasionally ignite because they're more often in range to use it. Not that ADC's can't, but you generally don't want to wait to be in range of an enemy just to exhaust them when you're low on health. Heal helps create more of that distance.
Anyway, back to exhaust's utility. It's much more valuable especially on certain supports, such as Leona, who can afford to be in a certain range or use it in tandem with their kit's cc against the enemy team. As the ADC, your job is to be safe above all else, because you can't deal damage if you're dead. Exhaust can be used offensively and defensively, and supports tend to hand you kills on a silver platter so it's generally better to let them handle it.
But that doesn't mean you can't, though. Go ahead and experiment. Graves is one ADC who might benefit with exhaust or ignite. Caitlyn probably would benefit more from heal. It all depends.
When I duo queue, I sometimes switch it up. Works well for me.
I think the main reason is that it's different, and people don't change unless they see it on a pro's stream or LCS. If they do it in LCS, watch it start appearing in solo queue within a week.
When I'm with friends, I (ADC) take ignite, and whoever is playing support takes heal. We are in a call together, so I can rely on them to heal me, and having ignite on the ADC helps to secure kills (particularly because I play Quinn ADC, who is very assassin-y, so the damage helps me burst even harder).
Probably because it used to be Support Exhaust ADC Barrier
Then when heal got buffed we switched to heal and never really worried about it after that.