There are two ways of playing audio CDs. In the old fashioned way, the drive plays directly to the audio connector and/or the headphone jack on the front. If you're playing the CD that way, software on your computer doesn't really matter - for as long as the drive receives the instruction to start playing, that's it. And actually, there were some drives that had buttons to start playing audio CDs on their own, so if you had one of those, you didn't even need the software.
Eventually, as CD-ROM drives started to be able to do digital audio extraction, that way got replaced by the more modern way where effectively, the software on your computer rips the audio CD through the drive and then plays it through the normal software stack. Trying to remember when that first became a thing; I think by Windows XP, that had become the standard approach, the internal audio connector and the headphone jack vanished (or, in the case of the internal audio connector, stopped being used - I think most motherboards with onboard audio for a number of years in the 2000s continued to have the connector), and that was that.
I would suggest that you need to figure out which of the two approaches you are trying to do. My guess is that Winamp is probably trying to do the second kind; it's new enough, not to mention has built-in ripping capabilities, so... that'd be my guess.