While the stated purpose of the rule that "only the winner
keeps the cash" is to increase the suspense going into Final
Jeopardy, I think there's an unstated reason: the show has
a budget. Imagine, if you will, this scenario (to use someone
else's example on this thread): first place has $10,000 going
into Final Jeopardy; second place, $9500; third place, $9000,
and no one wagers anything in order to keep the money won
(as might be the case on the original show). Total payout
for one day: $28,500. And since the show's on five days a
week, imagine similar scenarios every day. Conceivably, the
show could end up paying out $150,000 or more a week,
and I'm not sure the budget can handle that. The modification
that allows second place $2000 and third place $1000 is basically
to allow them to pay for their expenses in coming to LA for the show,
since many contestants are from somewhere else.
It's analogous to the original "Twenty-One." NBC allotted Jack Barry
and Dan Enright $10,000 for prize money; anything over that came out
of their pockets. Since they weren't about to part with their own money,
Enright and Al Freedman would arrange a series of ties and let the money
allotted by the network mount up (but then award only part of it, with the
rest--usually at least $25,000--going into Enright's bank account).