I was thinking that since XVA is on uncollaterized exposure, we should be using LIBOR discounting environment. Why don't we do that?
Libor, besides its name, is not a good proxy for uncollateralised funding. In fact, a large bank I worked for had an uncollateralised funding curve which was a a spread of +100bps to Libor.
The fact that it was a spread to Libor was for legacy reasons and easier to adopt to existing systems. However, a spread of say +115bps to CSA was equally appropriate.
When you trade a collateralised (CSA) versus uncollateralised trade and want to hedge there is an element of how you account for the trades on balance sheet.
If you mark as a (generally fixed) uncollateralised spread to Libor then you will also need to hedge the LIBOR/CSA basis. If you mark as a (generally fixed) uncollateralised spread to CSA then you can effectively ignore the LIBOR part of discounting completely.
A second point is that to provide an effective comparative base every derivative should be first valued with the CSA curve and then the adjustments (CVA/FVA/KVA) measured from that basis, this avoids any kind of double counting, and provides a standardised means of comparison.