The IRS budget has been slashed significantly in recent years. While sending out the letters would be cheap, investigating when taxpayers reply would not.
Prepaying property taxes in December has been common practice for many years, whenever a taxpayer anticipates that their tax situation is going to change in the new year, so I expect the IRS already has procedures in place for this. The only difference now is that there's likely to be a big upsurge because the tax bill affects so many people. But the IRS didn't have much warning, and they didn't get any kind of budget increase, so it seems unlikely that they'd be able to do anything to try to take advantage of people not having legitimate documentation of the transmission date.