A foam surround is not part of the radiating area, all it does is minimize reflections back into the cone, center it, and provide an air tight seal, that's it
The surround has nothing to do with the driver's T/S parameters beyond Xmax
A driver relies on the spider, the coil and the mass and profile of the cone for the low frequencies and mid band response, that plus the gap and box volume
Quality replacement surrounds like Rick Cobb's are always the best way to go and give you the best shot at getting things as close as possible to back where they belong specs wise, but all the cheap stuff that's close enough can or might alter is the Xmax of the driver, not create distortion
Meaning if it fits dimension wise
Changing the Xmax of the driver could change the driver's frequency response without a doubt, but if you do a refoam job with cheapie surrounds and get any other sort of audible distortion, and not just a change in response, then something else is wrong
It's not the cheap surround's fault