By the time of the original trilogy, [the Alliance is] pretty much a proper military organization, but it makes sense that it didn't start out that way.
The Essential Guide to Warfare touched on this as well, stating that any Confederate remnants that lasted past 23 GR/12 BBY (the final mop-up in the Legends timeline, though most of the major holdouts were accounted for in the Empire's first two years) were treated very much at arm's length by Mothma and Organa (who had been Republic loyalists), to say nothing of fellow Senator Bel Iblis (Corellia had largely sat out the war, to the detriment of CEC afterwards -- though Corellian yards did, as we saw in
Solo, build Star Destroyers under KDY license: and those that came out of said yards were just a little bit faster in real- and hyperspace (explaining the other odd line said by a certain charter pilot in
A New Hope.)).
They (Mothma, Organa, and Bel Iblis) didn't want to give an easy win to Imperial propagandists who, as it turned out, accused Rebel leadership of wanting to drag the galaxy back to the chaos of the Clone Wars anyway.
The Guide also mentions a CIS naval commander who, while not allowed to join the Alliance Navy, was given a letter of marque to harass Imperial shipping on their behalf....