I think that is the right thing to do unless you think the buyer will gut the place or may have plans to install a tankless. Nothing says you can't call the buyer or have your agent call their agent and find out if they have any such plans. Let them know you will replace it (your expense but not upgrading) so they don't have cold showers in this winter weather and in hopes that everything will go smoothly for both parties. Doesn't hurt to let them know you are doing them a favor. Hope that will provide the good mojo to get everything else finished smoothly.

And BTW, this would normally be a bite the buyer when the move situation since they didn't request an inspection. But things can fail unexpectedly even then. We've always had inspections and always found some surprises -- sooner or later.

Most RE contracts will say that the property must be delivered to the seller in the same or better condition as on the day of signing...

Hot water tanks usually have great warranties. Note the serial number and date and call the manufacturer.

Don't know if this is too late but I just saw the post.

One option is to purchase, at your expense and for the benefit of the buyer, a home warranty. Yes there is a deductible, but the warranty should cost about the same as a water heater.

So they are covered for much more than the water heater (HVAC, appliances, etc), usually for at least one year. They will be happy, and you will save time and hassle.