My point is that when I try to install the drivers for the WM8960, I get kernel header issues on the older version of the kernel:...

The advice to update to the latest kernel/firmware was so that you don't have to install any drivers and just configure the HAT's overlay in config.txt. Doesn't that work ?

I’ve asked the software developers, but they keep telling me to upgrade to 5.10 kernel, which RetroPie seems to not work with for me anyway.

RetroPie always uses the same kernel as Raspberry Pi OS. Version 5.10 has been available for quite some time now and there's no inherent incompatibility or problems running it with RetroPie (well, except maybe this issue, but that shouldn't lead to a crash in RetroPie).

Have you experienced any issues after updating to 5.10, or can we expect any update to the kernel any time soon for RetroPie?

No, I haven't experienced any issues with 5.10. As said above, RetroPie uses the Raspberry Pi OS (nee Raspbian) kernel/firmware, there's no separate RetroPie kernel update.