A stick pump is inserted between the spray skirt elastic and your body, so you don't have to open the spray skirt. Most of us store the stick-pump under the coaming, so one has to open the skirt to gain access. Usually if you've just done a reentry, the skirt is off anyway. One can also reach down inside the skirt elastic to get the pump. If you picture this, you likely to also picture flipping over with your hand down your skirt. (Does wearing skirts make us kayak guys all transvestites? That's one taunt surfers haven't thought of yet.)

A T-rescue does indeed dump out almost all the water. Enough remains, an inch or so that, sloshing around decreases stability, and if chilly, uncomfortable. My experience is based on my Great Auk-with a modified deck, recessed coaming --so I might have more of a problem than most kayaks.

The rechargeable battery idea is interesting. Most electric pump solutions use re-chargables that are rectangular and waterproof.
My experience with cordless drills (and based on my power tool repair guys larger experience) even the lithium batteries go bad after a year or so.
I've got an old Makita 9 volt, a newer 12 v, and an even newer 18v. The only one that is still useful is the 9 volt. The others not used for months at a time seem to kill the batteries.

My tool guy said if you were to test them as most people use them, the 9 volt will turn out to be most useful. The higher voltages allow for a lot of torque, but exhaust their charge quickly.
If one is just drilling a bunch of 1/4" holes, the 9 v will prove to be more useful. I think he said it was like using an huge truck to move a few 2x4s.

For kayak use I'd say start with the cheapest, lowest voltage lithium battery and design the system around that. (for me I'd use the Makita 9 volt, use them in my drill, buy a new one every year or two). It'd be difficult to seal the connection. I know my rechargeable battery on my waterproof radio is corroded in place.