Answer to OP's original question? No. Not how it works.
People who like to mention "full faith & credit" might want to read more of its background and why it exists in the first place.
People who think this is just like driver's licenses might find the history of the Driver License Compact, the Non-Resident Violator Compact and the Driver License Agreement interesting (and not what they might expect, or assumed). It was a long and difficult path to finally get the driver licensing departments of the states to reach some consensus. That was at the state level, too, not congress.
Then, look at what it eventually took to get
congress to accept and pass the federal legislation that became LEOSA.
I could easily see congress take the simple and easy step of looking at LEOSA as some sort of framework to develop a "national reciprocity" legislation for private citizen licensees ... meaning including some "common sense" things like requiring a similar sort of annual training and qualification (to the licensee's resident state standard, or an acceptable equivalent).
Sound good to you? After all,
that was considered necessary by congress (meaning in order to satisfy the concerns of some states) for active and honorably retired
cops, meaning men and women who had
already received a good bit of training in use of force, laws, firearms training, etc.
What do you think those same states (members of congress and the senate) might think would be necessary for private citizens, who don't generally receive all the same original and continuing in-service training and legal updates as cops?
I'd not be surprised if they decided that at least the same annual qualification "requirement" might be considered necessary for private citizens in order to carry concealed weapons among all the states, as well as the other restrictions contained within LEOSA. (If not additional ones.)
I'm sure all the citizens with CCW/CHL licenses from various states would be good with being required to qualify annually, as cops do for LEOSA, right?
Just look at how some of the existing reciprocity agreements have sometimes changed because of differing state opinions about what qualifies as "sufficient" training, physical class attendance and range quals. Want them to do it all in some one-shot deal for fed level legislation? I'd not be surprised if it didn't end up
more restrictive, overall, than
less restrictive overall.
Careful what you wish for ... because it probably won't be what everyone expects.
Probably hear no shortage of things from politicians like, "This is common sense legislation", "It's not perfect, but it's the best we could get passed", etc, etc.