Pain on bottom foot when freewheeling for a time in a gravel descent

Ligaments hold all your organs in place, you may find that the nerve ending is on the receiving end of those ligaments. A basic way to look at it, the boys in ww2 on the landing craft would get terrible stitches and vomit not from sea sickness but from there organs being held in place while there bodies and knees would be bounced around in the rough sea. You may be having mechanical stress and feeling pain from that. Just a no medical persons look at transfer of force.

Leftaroundabout nails it - in addition this sounds like a time when technicque needs to be updated.

You might try ghost pedalling, where you turn the cranks over but not fast enough to push the bike.

Another idea might be to descend like a racer, where your weight is spread between two feet and some weight on the hands, and almost no weight is on your sit bones. Instead you have the insides of your thighs pressing on the sides of the saddle nose, so you can push the bike around under you. Your knees and elbows should be slightly bent and actuate to absorb minor variations in the ground without your torso's path changing. IE your limbs are your suspension.

Counterintuitively, your descent might be easier if you vary the speed a bit. Go a bit faster and you may smooth the cobbles, at the risk of loosing traction. If nothing else, the descent will take less time.

Also try looking at your braking technique. Brake harder for shorter duration, and then release. You can limit your speed by sitting up and letting your torso front catch more air, and square your shoulders to be wider.

Inserting standard disclaimer about accepting medical advice from the Internet; this is based largely on what I learned from my physiotherapist when rehabbing after an accident.

Replying strictly to your question about pain in the foot that is held in the down position, it seems reasonable to hypothesize that you're not making a conscious effort to rest your weight on your toes. If you were doing that you'd almost definitely feel fatigue in your calf muscles before pain in the bottom of your feet.

You can check this with a simple experiment: try standing in bare feet and lifting your heels a centimetre off the floor and holding it. After a minute or two if you don't start to feel the same discomfort then it's very likely caused by your foot position. (Of course there's a good chance you'll start to feel fatigue in your calves. :) You can double-check by standing with your toes on the edge of a stair, and lowering your heels to level with, or slightly below, your toes. If you do start to feel that discomfort in this position, it's definitely your foot position.

As to solving the problem, Criggie's answer is best; ghost pedalling will keep all your muscles engaged and rotating through the full motion will keep you from hanging your weight from your toes. But I understand that to help with control you may want to be in a stable position, feet braced, in uncertain terrain. In that case standing on horizontal pedals and concentrating on keeping your weight on your toes is better than standing on one pedal. (Switch which pedal is forward every time you turn or brake, in order to keep from getting fatigued.)

The other problem with keeping one foot down is that you're putting much more than 50% of your weight on that side, which will exacerbate the stress on that foot.

Sometimes I inadvertently try to "grab" the pedal with my foot. This usually happens when riding with flat pedals and when I am insecure. For example, spinning high cadence, slippy pedals, wearing flip flops, and scary descents. This grabbing leads to fatigue, muscle pain and cramps.

Maybe something similar is the source of your pain. Since you already have a reliable pedal-foot interface what you can improve is being secure in your bike. That you have one pedal low very much hints at this.

Follow Criggies advise and learn to balance on both your pedals at level cranks. That is the most stable position to take bumps and approach difficult terrain. You can easily practice the stance on safe level ground going slow.