Just posting an update on these RTC's. I have been testing 6 different units (from 2 suppliers) and they seem to be working Ok, including the 32k I2C eeprom. I did some modifications to them: First was to de-solder the power led, and the second was to remove the charging circuit (by desoldering one resistor) , and replace the rechargeable lir2032 button cell with a non rechargeable cr2032.
I have taken them down to 2.8 volts, and not yet seen any problems with the RTC, or read/writes to the 32k eeprom, even though the units original spec was for 4-5 volts. More tests are ongoing, but so far they seem to be working. Of course, at this price, I am still a bit uncertain of the quality of that DS3231. Lots of talk on the web of ghost runs, or people grabbing out of spec chips, that were supposed to be destroyed, and sending them back to market in these discount boards from China...