The same way you prove cost basis; if you have adequate records of one, you’ll pretty invariably have both.
If lacking transfer documents makes the sale value “all gains”, then the inflation adjustment is irrelevant, since you are adjusting a zero basis value, which is still going to be zero if you do or don't adjust it for inflation, no matter what a amount of inflation has occurred.
The only place where there is even a theoretical problem is where there is proof of basis value but not basis date; this seems extremely improbable to occur in practice.
So you already need a receipt or some record with the original purchase price, which probably has a time stamp on it.
But yeah this is one reason why collectibles have a special, higher LTCG rate than securities.
How are electronic records relevant? The information needed is expected, whether records or electronic or not, when doing capital gains now, so there's non rational need for electronic records.
> For other types of assets, the inflation rationale is still a decent reason to have a lower LTCG rate
No, it's not. For the reasons already explained.