Nope, you don't need the original hard drive that the computer came with. There is no legal work-around for this. Windows 7[.1] does not upgrade directly from Windows XP. A recovery/re-install disc isn't going to work. Also, there is no Windows 7.1, there is a Windows 8.1 though. I recommend you contact either Microsoft or whoever you bought that disc from and hash it out with them. A lot of Windows 7 and 8/8.1 activation codes are not printed on the machine but are hard coded into the computer's hardware itself so you may need to buy a legitimate copy of Windows 7. Don't blame the lack of functionality on the computer here. It's just telling you that you can't do what you're trying to do.
I was doing a clean install, not upgrading directly from XP.
A search on Google for Windows 7.1 (Windows 7, service pack 1) returned more than 64,000,000 hits.
The 7.1 disk is a legitimate reinstallation disk issued by Dell for use with Dell machines. Perhaps the problem is that the 755 originally shipped with Vista 32 or 64 bit software, and the product key won't work with anything later.
I wasn't blaming the lack of functionality on the machine. I am simply somewhat frustrated that my machine now lacks full functionality, whatever the reason. If in the end I'll have to buy the full install version of 64-bit 7.1, so be it, but I want to make sure that's the only workable solution before I spend the money.
At least the issue about the hard drive is resolved. Thank you for that.