Why is ascending some stairs more exhausting than descending?

However, when descending, you have to exert an equivalent force to stop yourself from accelerating and hitting the ground...

Absolutely correct.

So is the fact that ascending stairs is commonly perceived as significantly more exhausting than descending the same stairs purely a biomechanical thing, e.g. having joints instead of muscles absorb/counteract kinetic energy?

Right. When going up the stairs, you must exert large forces by your large muscles. When your legs raise your torso, your muscles are supplying sufficient forces (with an energy cost) to do so.

When you go down the stairs, it is not the reverse of ascending. Instead of using your large muscles to decelerate, most people will take a straightened leg and plant it on the lower step. The deceleration is accomplished by plastic deformation in joints, fluid displacement in your foot, and the materials in your shoes and the floor. There is still some energy demand on the muscles for coordination and moving the legs, but it is significantly less than if the muscles were doing the deceleration job.

you have to exert an equivalent force to stop yourself from accelerating and hitting the ground

As animals, we spend calories to go up and gain potential energy. Tiredness is a measure of spent calories. Ideally going down needs no calories, and we have not evolved to the point of taking them back. Except a few calories are needed in the interaction with frictional forces and some skill to transfer the excess energy to the steps.

Think of skiing. To get up the hill on foot needs a lot of calories, (believe it or not back in 1958 I was taught to walk up with skis) to control the speed sliding down a few, and some skill (that is why at that course, I was fine going up, but ended in a splat at the bottom of the hill, having no skills). The energy is the return of the calories spent to go up (well partially, friction takes up a part of it).

Edit after edit of question:

The only reason I introduced velocity into the question was to show that you actually have to expend energy going downstairs

You are ab initio assuming that the velocity takes energy from your muscles. The going down velocity is sustained by the diminishing of the potential energy incrementally by going down a step. That turns into a velocity of your body, hitting the step a normal force bounces a ball back, you have to spend some muscle energy in order not to bounce, but in no way equal to the energy needed to carry your weight up one step.

I am quite positive that friction does not play a significant role in this thought experiment.

Wrong. Friction plays a very significant role in walking, climbing up or down. Have you tried walking on ice?

No, I am not arguing that descending is subjectively less exhausting, I am asking why it is less exhausting

It is less exhausting because less energy is needed from the muscles of the body, needed in directing the way of descent to control the release of energy from the incremental lowering of the potential energy of the body. Directing is much less energy absorbing than lifting.

There is no "free" or "automatic" normal force emanating from the stairs that stop you from accelerating.

You paid for it going up the stairs. The incremental velocity of lowering the body a step at a time hits the step and a normal force is created from the impact, not from the muscles. The muscles need to control against it so you do not bounce like a ball, but that is less energy than the potential step, because of friction taking up most of it.

Also, as several people pointed out, we as humans have no way of using or reconverting our stored potential energy to decelerate ourselves.

No, but our body is smart enough when in a velocity situation to spend a bit of muscle energy to direct where that velocity goes. The velocity coming from the acceleration of falling from step to step is transformed to friction (no slide shoes help) and a bounce of the body due to the normal force, all eaten up in friction and radiation. The new energy input is small with respect to the energy spent to get at a high potential. See the ski example above.

After the third edit, here is a simple example:

1) Take a half inflated ball that would bounce a few times and stop on a flat floor.

2) Lift it upstairs, next to the edge. Potential energy acquired.

3) Give it a small push just to fall on the next step: a tiny bit of energy expended.

It will bounce down the steps without any extra energy and, depending on how deflated it is, may reach the ground, or stop in between due to the normal force being larger than the gain of kinetic from potential energy from the fall of one step.

• Your muscles exert more force when ascending than descending:

When going downstairs they have to exert a force smaller than gravity's in order to control your speed, while when going upstairs, the force they exert must be at least equal to your weight, in order for you to ascend. So your muscles are doing more work ascending than descending, the movements are typically not symmetrical.

That's especially true because the braking force (for the "fall" from step to step, provided by the step's normal force) is not a reaction from a force exerted by your leg muscles - you can hit the step straight-legged, and let the impact energy dissipate passively through your body, spending very little energy in the process, as well explained in BowOfRed answer.

• The natural energy losses help you keeping a comfortable speed going downstairs, while that's a loss you have to compensate for when going upstairs.

• And, yes, there are also certainly some biomechanical aspects at play too. Consider, e.g., how much more tiresome it is to descend in slow motion: going downstairs very slowly is hardly any easier than going upstairs at the same speed - it increases the symmetry between both movements.