I like the idea of it, in that it gives you the option of customizing your subclass to fit whatever class flavor you want, without sacrificing the core progression of your base class. The fact that you can still get your spell levels, or extra attack, etc. on time, for example, while picking up some features from other classes is nice. Compared to RAW multiclassing, I would characterize this as giving you more "horizontal" flexibility, by effectively giving everybody 11 new subclasses they can choose from.
You sacrifice "vertical" flexibility, though, in that you don't have as much freedom to adjust the proportions of each class in your build: you have a primary and secondary, and they progress in a fixed way. For example, if your primary class is fighter or rogue, and you want to get a little bit of wizard, your only choice in this system is to take, essentially, what amounts to 8 levels in the RAW level system. That makes you sort of a 1/3 caster, like an eldritch knight or arcane trickster; and that's the only way to do it. If you want a more even split between martial and caster, you can't do it.
As a player of an arcane trickster who chose to multiclass into wizard because I wanted more than the 1/3 caster that AT gives you on its own, this system doesn't address my issue with the existing system, which is that there's no way to sort of titrate your build to accelerate your magical progression just a little at the cost of just a little delay in your primary class's progression: you either wind up just flat out delaying your magical progression by dipping caster, or you have to invest a lot of levels, thus paying a lot on your primary class side, to get a little bit faster progression of your casting. What I would have liked to see is a way that you can take one or two wizard levels as an EK or AT, and have this allow you to gain 2nd / 3rd level spells a little bit sooner than a single classed EK/AT, while delaying your fighter/rogue features by 1-2 levels. However, this is arguably a bit of a corner case, since EK and AT are the only single classed (full or part) casters that don't have their own spell list.
As far as balance goes, my chief concern with your multiclassing system is that, having collapsed multiple levels into "level 1" of a class, you're getting two levels worth of stuff for each level in your secondary class, but giving up only one level worth of stuff by trading away the corresponding subclass features. If it were "first set of subclass features" vs "level 1 of another class" in the vanilla level chart that would be more balanced, but the collapsed class chart skews things in favor of muticlassing, it seems to me.