Perhaps, if you do want to use Exhaustion in this way, you could offer some way of mitigating it, but only a few times a day, perhaps something like:
Recuperation
Exhaustion is a serious, potentially life-threatening problem, so it's natural for characters to try to do something about it. However, there are limits to what a physical body can achieve, even with the aid of magic. When intense fatigue sets in, there's only so much one can do to offset it before needing to rest, and the harder you push yourself, the longer it can take to recover. This optional rule helps mitigate some of the concerns with Exhaustion while still ensuring that it has bite.
Variant Rule: During a short rest, someone proficient with a Healer's Kit can provide recuperation to a number of creatures equal to their proficiency bonus, consuming one use of the kit per creature treated. This recuperation removes one level of exhaustion. No creature can benefit from recuperation more times than their Constitution modifier, down to a minimum of once. Unfortunately, adventuring out in the wild is hard, and slows one's ability to bounce back from difficulty. A long rest allows a creature to benefit from recuperation one additional time, up to their normal limit. If the creature taking the long rest would regain hit dice beyond their maximum capacity, each hit die beyond their maximum instead raises the number of times the creature can benefit from recuperation, up to their normal limit (Constitution modifier, minimum once).
If you wish to encourage the use of magical healing, you might allow lesser restoration and other such spells to work as a substitute for a healer's kit. You may also consider granting proficiency with a healer's kit to certain classes or subclasses so your party can make use of this feature.