The reasons GBAS and SBAS are the way they are goes back to decisions made in the early 1990's when the concepts were first proposed. When the push started for satnav approaches it was obvious that augmentation was necessary to meet the accuracy and integrity needed for approach procedures. The FAA tasked RTCA to develop the standards and that was delegated to Special Committee 159 (SC-159).
SC-159 is a very large committee with representatives from government agencies, airlines, aircraft manufacturers, general aviation, avionics suppliers, and pretty much anyone with an interest in the system. Because of the huge amount of work and the differing systems the work was broken out to separate subcommittees. So while there were some people working on both subcommittees, the subcommittees each took there own path.
GBAS was being promoted by the FAA to be a functional replacement to ILS with support for CAT I and eventually CAT II/III operations. This needed support of the airlines who were only willing to go along if it didn't add significantly to their costs (preferably it would provide capability where ILS wasn't feasible and give them additionally capability at little to no cost.) So a fundamental goal of GLS was to be as operationally similar to ILS as it could get - minimal crew training, minimal aircraft system changes.
That last bit is key as the ILS is a DAL A system tied to the autoland system which is very expensive to certify. The avionics solution to this is the multi-mode receiver (MMR), ref: ARINC 755. The MMR allowed for retrofit as it occupied the same package as the legacy ILS receivers.
The MMR added a GPS sensor (which wasn't standard in the early 90's) and it allowed for integration with the ILS and its interfaces. The first MMRs were just and ILS and GPS sharing a box. But later versions added a precision approach navigator (PAN). The PAN could use the ILS output busses to provide the "ILS Look-Alike" output. But the GPS needs the augmentation data. This is broadcast from the GLS ground station on a VOR/ILS frequency. So the ILS receiver (a software defined radio) is repurposed in GLS mode to receive and decode the VHF data broadcast (VDB) from the ground station. It can then provide the augmentation data to the GPS and the FAS data to the PAN.
The reason for the FAS data being broadcast and not being in the nav database is that database is located in the FMS, not the MMR and the FMS is not a DAL A system. It also follows the concept that the ILS/GLS is a stand-alone unit that can support a precision approach even if other systems have failed.
Conversely, SBAS was pretty much ignored by the airlines (they only added WAAS GPS receivers when ADS-B drove the need.) SBAS was championed by the general aviation community that want RNAV approaches into airports that didn't have good approach options - no ILS, maybe a VOR or NDB approach. An RNAV approach was a big step up. Due to the large number of small airports that desired the capability, SBAS was more cost-effective than a GBAS approach. It was also cheaper for the user as it was contained within the GPS navigator/FMS. The limited bandwidth of SBAS forces the procedure data to be included in the nav database. But SBAS will likely never go beyond LPV capability which is acceptable to the GA community.
So the different needs of the GBAS and SBAS resulted in the different solutions.