I can't quite remember exactly when I felt confident in my party. I want to say that it was sometime around the port city with the pirates (Balfonheim). Like you point out, there are limits on the gambit system early in the game, particularly the lack of commands available and the limited number of commands per character. It took constant experimentation, especially with the prioritization of commands. Then again, I loved grinding so damn much in this game, that my character's higher levels and number of licenses may have attributed to my success.
One trick I did was diversifying which characters handled which status effects in order to cover a fuller range. One would use antidote or the equivalent magic for poison, while another would cure some other ailment, etc. Same for boosts, like bubble, protect and haste. To a limited extent, I was able to do the same with attack magic: one character may use fire magic when coming across fire-weak enemies, another would use lightning magic, etc. Same with steal and other techniques.
Prioritizing restoration commands took the most effort. I ended up using (imaginary) jobs to my advantage. One character would be primary support, and would be the first to stop attacking and start curing. Simple restoration at first: for example (percentages may not be exact) <70% would be cure, <50% cura, <30% curaga, <20% curaja. Another character may start jumping in at <50%, and finally the last will join in to help at <30%. During emergencies, everybody stops and cures each other.
Naturally, attacking enemies is lower on the list of priorities (don't attack until defenses are secured). However, I like to keep my tank character on the offensive until emergencies. He'll have the most diverse attack commands, and will use powerful items like x-potion for a quick, effective recovery when necessary.
It's been years since I've actually played the game though. I can give you a better answer once I get through the re-release a little bit, and granted that the zodiac job system doesn't screw up my method.
EDIT: for some reason, PS4 downloads have never been quick for me. FFXII has been downloading all day, and only requires a 1/4 of the data in order to start, and now that it's past midnight on the east coast and the game is officially released, my console is still only halfway through the 1/4 of the data necessary to start.

Guess I'm waiting until tomorrow.