Tires with foam liners have been around for almost a decade. The procedure is well known.I got a flat the first week with 400 miles on the clock. Couldn’t find the leak until I removed the tire. Nail was gone but you could still see the impression of the head and the tiny hole. Took it to tire shop and they called and said they patched it but they couldn’t guarantee the repair because the goo holding the foam might not be compatible with the stuff on the patch. They showed me pictures of the repair and didn’t replace the small section of foam. About 2,000 miles later it’s holding fine. Also found a YouTube video by a company that showed proper materials and techniques for patching foam lined tires. In my case I believe a plug would have worked just fine.
Your shop should have replaced the square of foam they removed to repair your tire. There's nothing magical about the "goo" holding the foam.