OEM product keys are not compatible with other distributions of Windows (Retail, VL, etc.), and OEM product keys are not compatible with other OEM media (generic MS, Dell, HP, etc.), as the keys are OEM-specific.  So, for the same reason as not being able to use a Dell disk on a non-Dell PC, you cannot use a non-Dell disk on a Dell PC.  That said, we are talking about the "automatic" activation tool ... when you activate by phone, it translates the Dell OEM key you enter into a validation code, which you will give to Microsoft; if it checks out, then MS will give you an activation code that will allow your Windows to activate.  Calling in to activate over the phone is the only method of activation when using mismatched media/key.
I recently picked up a used Dell and although it had XP Pro installed the sticker on the case is for XP Home, so in order to be legit I installed XP Home. Unfortunately I couldn't locate my copy of Dell XP Home at the time so I used a generic OEM copy, but I have another Dell machine with XP Home and I probably saved the volume key from the Dell disc in a text file before when I used nLite to slipstream a service pack since it displays it. So I know you can't use a Dell disc on a non-Dell machine, but will a non-Dell copy of XP work if I use the MS Key Update Tool with the volume license key? I think the one on my Dimension B110 may have even been the same as the one on the disc when I used a keyfinder program to look at it.  Sorry if this is too simple/obvious, I actually did search for the answer.
Can I use a non-Dell XP disc if I use the key found on a Dell disc/install?