I'm sure they'll just throw error messages until they've managed to establish a connection. Device manufacturers have already shown they'd rather cripple your devices than respect your wishes.
Antennas on cheap consumer devices are often just a trace on the PC board. Still not that hard to remove (could just cut the trace), but harder to do without damaging anything than unplugging an external one.
Seems like an interesting cottage industry that could bloom around intentionally stupifying your unwanted smart home.