It appears the flight ID is not sent in a Mode-S all-call DF11 response, only the ICAO code.
A short-form DF 5 Surveillance Identity Response does seem to include the flight ID, but not sure.
EDIT: I checked and only the Squawk (encoded in the 13-bit ID field as 4 octal digits) and ICAO address/parity in the AP field are sent in DF5 - No alphanumeric pilot-entered flight ID.
The long-form DF21 Comm B Identity Response definitely has the flight id, as you mentioned.
Around Chicago, the rate of DF21 messages is very, very low, at least as received by my station, which has an excellent antenna and receiver system. DF5s are quite common though.
I seem to get about 20% of the Piaware MLAT flights populated with flight number from the local receiver in Planeplotter, but it varies wildly.
I doubt I ever pick up TIS-B broadcasts directly, but I do occasionally receive them when propagation to a ground station is favorable.