Next week, flying AA ATL-CLT-MAD and IB MAD-TLV all on same biz class award ticket with an 8 hour layover in MAD. Will check luggage to TLV. I have both US and EU (UK) passports. Arrive MAD around 8:30am. Was thinking I would enter Spain using EU passport (I assume that's quicker), store my backpack in T4, take the Express bus into town (as the Cercenias train is not going all the way to Atocha currently), take a self-guided walking tour past the Prado, fountains, Puerta Del Sol, Plaza Mayor, grab a bite at the Mercado and walk to see the outside of the Palace and gardens. Then get an Uber back to the airport around 1pm for my 4:10pm flight to TLV and exit Spain through immigration using my EU passport again. Hopefully that will give me enough time to shower/get a drink at the T4S lounge. Doable?
Definitely doable. As you exit at arrivals in T4, turn left an go all the way to the end to store your bags. Alternatively you *MAY* be able to store them at the lounge (which would involve an extra security check and some backtracking) and you can also obviously store the bag at Atocha.
As you rightly pointed out, I would NOT recommend taking cercanias from Madrid, but if the schedule is right, take it from the airport - there are enough trains between Chamartin and Sol.