On the other hand, if I were hit by a loaded shopping cart at 3 mph and I weren’t allowed to move my feet, I’d probably be okay if I saw it coming.
Keep the same momentum but scale the mass down to, say, 200g. Now it’s supersonic, and it will hit with maybe 10x the energy of a musket ball. This would be extremely damaging and likely lethal.
(If you make the mass too small, it might go straight through a person, in which case not all of the energy is delivered.)
But the point was that if you were hit with a full container ship at 0.1 mph and not allowed to move you'd be dead.
The only reason why a high energy but slow moving object (thus high momentum) is harmless is that it will deliver a small acceleration to you and you won't suffer from the consequences of a high acceleration (which is what generally kills people when they hit something like the ground at high speeds). But that assumes that you're free to be accelerated and move away.
I think this is a bit too simple. People can get injured and objects destroyed in multiple ways. For example:
One could be torn apart. If you are glued to the ground and you are pushed away with too much force to resist, you have a problem. You can stretch or move a certain distance and withstand a certain force while doing so, and the product of those numbers is the energy you can remove from the thing hitting you. If the impactor is at a fixed speed, then you can turn that momentum into an energy, but that’s a bit silly. You really can stand still and deliver more stopping impulse to an object moving toward you when it’s moving slower because you can resist for a longer time.
One could collide with an effectively infinitely massive object. But the momentum of that object seems irrelevant — hitting a brick wall on wheels at 25 mph seems much worse than hitting the Earth at 0.1mph despite the Earth having many orders of magnitude more momentum. The impact velocity seems more relevant.
One could get hit by a fast-moving projectile. This is complicated, and neither momentum nor energy seem like sufficient measures. As an extreme example, ultra-high-energy cosmic rays surely hit people on a somewhat regular basis, and I’ve never heard of anyone noticing such an event, despite the energy involved being quite macroscopic.
(This is about the estimated yield of the Beirut explosion.)
I would rather get hit by a slow moving object than a fast one with equal momentum.