Freewheel 6 or 7 speed wheels typically have no clearance problems there. The issue is likely being caused by rear derailleur hanger misalignment and/or the low limit screw being adjusted too far out. There are many questions about both.
Update post-image: The derailleur hanger is bent. Put the shifter in high position and poke a 5mm 3-way wrench or L-wrench into the pivot bolt of the new derailleur. Reef it around to make the plane of the pulley centerlines appear parallel with the plane of the cogs. Re-adjust the high limit if needed, double check it looks parallel, and you should be good. Eyeballing like this isn't great for latte speed generation bikes, but it's fine for 5 and 6 speed. If you want to baby the frame you could have it done with a proper tool, which I'd recommend in most other non-emergency cases.