All large steam boilers did and do pre heat the incoming water with exhaust gas. Starting back in the 17 or 1800's. I think they were patented as "steam economizers". But they were cast or ductile iron, and large structures. But could raise the efficiency to very high levels.
Your first problem is you might have double wall exhaust pipe, so of course you have to change to single wall.
You are NOT going to have a reverse flow of air, because the exhaust is not that hot [measured it?] and you wont extract but a percentage.
You could double wall the pipe and blow air in with a computer fan, and likely get the most for the least. Get a good CO2 detector.
If you had a power exhaust you could do as much as you like, even inside the pipe, and run it down hill so condensation doesnt go into the burner.
You might have a condensation issue on your standard pipe on the outside wrap too. If I wrapped a section, I would over sleeve it with a perhaps 10" pipe and fill it with vermiculite.
Perhaps cheaper is a new high efficiency heater. And if you are on natural gas, not propane, it probably won't pay back.
The main issue is that cheap gas water heaters are junk, and one heat exchange pipe up the center is a bad joke. Everyones is afraid of a spiral except Polaris. LIkely your pipe is hot because the base of your heater has 3" of calcium in it, and your efficiency is down to 20%.
When did you last REMOVE the drain valve and rod it out with a hose and hanger with water coming in the top? And adjust the burner gas flow level? Try to boil water on your stove with 3" of pea gravel on the pan bottom. Low burner level, unless you need the recovery, helps greatly with efficiency. A gas water heater with a large tank and about 6 pilot size burners inside would probably get to 95% efficiency.
We are in the DARK ages with standard gas water heating.
Polaris used to ship out their power vent, 95% heaters with a sort of "economizer", the air INLET 2" pipe ran through the 4" exhaust pipe for about 4 feet. I extended mine to about 20 feet, actually to gain some back pressure [poor ignition] and now the inlet air is fairly warm.
I feed my Propane Polaris with an electric 40 gallon heater, because when gas is over $2.75 a gallon, electric is cheaper, and you can
insulate the tank to r-38 in a moment.