That's your mistake for taking the car with tires issues to a dealership.
Find a local tire shop, and they will plug the tire without inventing lame excuses.
That's not true.
The "goop" residue would need to be cleaned from the patch area, but it's not hard. I've personally had multiple tires patched after on-the-road emergency sealer application.
The tire sealant does leave a bit of a mess on the rim and the TPMS sensor, and those will need to be cleaned up during installation of the new tires on the rim (when the time comes). That will be the time when whomever is installing the tires will utter a few unkind words about the person who had used tire sealant. ;-)
Tire sealant is designed to work itself into the area of the tire that is leaking air, and clog it.
The foam will be a minor barrier to the tire sealant doing its job. Since air is leaking past the foam, so will the sealant, and it will still seal the cut as long as it's not too large.
The bigger problem may come from uneven absorption of sealant into the foam around the circumference of the tire. That may compromise the balance of the tire, and introduce vibrations. Since you want to get the wheel to a competent tire shop to properly patch the tire anyway, it will be only a temporary problem.
HTH,
a