Steve Shantz wrote:

...My design would use parallel PEX loops to deposit and extract the heat.

From sand. Water seems better.

...How much mass can dense foam board hold before it compresses?

A lot... 30 psi?

The use of water for heat storage has several big advantages, one being
it's much higher heat capacity per cubic foot, but now the basement
floor must span the pool in the sub-basement.

That's one reason I suggested 12'x12'x4' tubs. You might fill a sub-basement
with sand, then add some water for greater heat capacity and easier transfer
(just pump water in an out, with no heat exchanger), leaving a layer of sand
on top which could support a floor, but 12' isn't a large floor span. Water
could increase fire safety.

Would the basement floor have to be concrete to resist the water just below?

I think it just needs a good vapor barrier. Plastic film and/or foamboard.
Putting the tank in the ground removes the need to build strong sidewalls.

What about when you get a leak? You aren't going to send a diver into 120 F
water to patch it!

A tub shouldn't leak, if carefully lined with a single 20'x20' piece of EPDM
rubber folded up like a Chinese takeout box. In the rare event that it does,
it might be pumped out for repair, which might just consist of adding another
layer of rubber. We could make 4'x19'x3' tubs with 10'x25' pieces of rubber.

If one shuts off the heat in April, what will be the source of cooling for
this insulated mass so that it is 55 =B0F come the 4th of July?

Trickle water over a north roof at night, or through a pond, under some rocks
for safety and shading. Or keep the tub hot to make DHW.

Nick