I think my issue is more with the number of techniques drilled per class and the randomness of what's done each week. I've seen a couple of discussions or videos where people describe a more structured approach at white belt. The one I've seen most common is that you teach a "system" or "combination" at a time. For example, a "system" might be how to break the guard, toreando pass into side control, transition to knee-on-belly, then mount, then go for an Americana. I use this example because that's a typical roll for me (on bottom).
Another approach I can think of, based on my TKD and HKD experience, would be to have a set of skills that white belts regularly drill. For example, a few times each of a couple of closed guard sweeps, closed guard submissions, closed guard passes, side control escapes, mount escapes, back escapes, and submissions from each position. It wouldn't leave much time for rolls, but it would get a set of foundations into long-term muscle memory faster in my opinion.
And I'm not saying do that forever. But maybe until 1st stripe.
My plan in my future TKD school is to take a similar approach - a set of techniques practiced consistently in high quantities at the foundation level, and then expand on that in a more modern "technique of the day" approach at the advanced level.