I think the situation as you've presented it is kind of silly. The economy in Sol is not just going to be "doing badly" so nobody has any money. It's going to be doing badly either Sol was relying on imports from other systems, or because it was relying on the money from exporting to other systems.
From command briefings during FS1, it's obvious that Sol was involved in a lot of research and development. That probably means a lot of military contracts providing funding from the central government in Sol. So the government is going to be missing a large chunk of the stuff it spent money on, but it also means that the economy as it pertains to high-tech equipment is going to be largely intact. I can also see that a lot of manufacturing and production plants would be based in Sol, possibly not the largest ones, but simply because they were there first and it's more cost-effective to scale them up than to move them to another system.
I suspect that one of the key components to be produced at those factories would be jump drives. This is the last key technology that Terran craft would require before they started colonizing other worlds, and it sounds as though improvements on jump drives were not revolutionary until the introduction of interstellar jump drives (and interstellar technology was perfected at Sol). Furthermore, the fact that the technology was moved from being invented to being mass produced so rapidly suggests a close association between the research team and the manufacturing plant, and that supply lines were not disrupted by the war (ie all critical points were located in secure systems). According to FS1 on 3/2/35,
This mission will be facilitated by the new subspace drive we have received. For years the GTA has tried to give a fighter the ability to do intersystem jumps. After monitoring the Beta Aquilae engagements, the GTA science colony at Sol has finally been able to solve the puzzle. All GTA fighters are currently being equipped with intersystem subspace drives.
The first mention of Beta Aquilae occurs in La Ruota Della Fortuna (2/15/35).
So my guess would be that jump drives are the most likely thing for Sol in the GTA to have a surplus of (besides people and government officials). Without a huge fleet to equip with interstellar jump drives but all the manufacturing capability, they'd probably be dirt cheap and easy to get hold of. Countless supply craft would've been diverted to Earth to evacuate the government and populace; the fact that the remnants of the GTA were in such turmoil suggests that no clear government existed and that most of it was trapped back in Sol. Thus those transports would still be around and would need something to do to remain profitable.
Finally, Sol would have all the vessels of the 1st Fleet hanging around with nothing to do. Sure, they could keep order, but that'd be a job for intrasystem police, not the GTA's elite fighting forces. So it'd be perfectly reasonable for large military transports to be diverted to ship civilian vessels across the system. Transit between planets should only take GTA vessels a matter of hours, and rescue would only be a matter of minutes or seconds away. Flying across the system would not nearly be as unrealistic as it is today; in fact it might be more analogous to taking a road trip across the United States, except emergency crews could actually get to you much sooner (unless you encountered a crisis during re-entry or something).
re: GTVA's technological ability
I doubt it as well...the Lucifer was pretty much vaporized; the back half was trapped in the subspace node and never emerged. The front half blew up and large chunks were apparently vaporized. Given that the jump drives and hangar bays were on the back half, I don't think it's very likely that Sol had any physical Shivan jump drives to study. Computers are a possibility, but I have real doubt of the GTA finding anything even more than slightly intact on the Lucifer. Given that the Knossos was apparently Ancient and not Shivan in origin, I don't think that the Lucifer even had the technology to do anything similar to what you're suggesting. It's possible that just having the more advanced technology might provide some clues, but it's really just wide-open speculation.
So if you're asking if it's a reasonable progression of events from Freespace 1, I would say no. If you're asking whether people who play a game with 2 km-long interstellar spacecraft that are limited to the maximum speed of a fast biker, and interstellar fighters that can't go faster than a Cessna, all of which operate according to principles that directly violate the laws of physics, would be willing to accept this idea? I would say probably.