Yet a cap-weighted index will have you buying more overvalued stocks and less undervalued stocks.
There is no evidence at all that this is correct. No evidence that the largest cap stocks are overvalued and the lowest cap stocks are undervalued.

Well, there is the theoretical evidence which shows why cap weight is a bad idea for good returns, and also the empirical evidence that it's an underperforming strategy which confirms the theory.
What sort of evidence were you looking for?

Sometimes a few very large firms do better than the average, as has happened recently and in the late 1990s. This is very much the exception, historically speaking.

A random (and amusing) paper which is a bit of a mix of theory and backwards looking empiricism.
researchaffiliates.com - P aug surprising alpha
The most interesting conclusion isn't that equal weight is good, nor that any other particular "alpha seeking" weight is good, but that cap weight is quite the outlier for poor performance among all the possible weightings they tried. Usually, and on average over time, but certainly not always.

e.g., for any random 12-month stretch 1986-2016, there was a 64% chance that a slate of the five largest stocks by market cap (rebuilt quarterly) would do worse than the S&P 500. The biggest 5 lost the race by 3.18%/year overall. This is not a proof that the biggest 5 will do worse in the next year of five years (though it's in general the likely bet), but it is a wonderful rebuttal to the notion that you obviously always want to overweight the largest because, well, they're the biggest winners and/or it's different this time.

Jim