What Igneococcus says is correct. Women have two X chromosomes in each cell, men have one X and one Y, but (pseudoautosomal region excluded) one X is inactivated in each cell.
A gene is a DNA sequence that codes for a protein so it’s debatable whether you can refer to a sequence on an inactivated chromosome as a ‘gene’ at all. The genes on the active X are coding for the proteins.
A man doesn’t lack DNA found on the X chromosome. If he did he couldn’t produce sperm cells that contain the X, and he couldn’t father a girl.
The default developmental phenotype is female. The SRY gene over-rides this default developmental route. The Y chromosome is not some feeble half-X; it is a pared down delivery mechanism for male sexual development.
It does not carry superfluous genes because if it did these would only pass down the male germ line.
It is not being degraded. It is being streamlined. Evolution isn’t going to eliminate it because natural selection acts upon mutations that can be passed down. A mutation such as the deletion of the SRY gene isn’t going to be selected for because the SRY gene is necessary for reproduction.
I have had this same conversation with TRAs who insist that intersex conditions that can cause infertility are ‘part of evolution.’