I don't like the idea very much. There are so many classes (12), that the number of combinations (132) would require an enormous amount of design work if each combination was done ad-hoc. Eventually this would mean that some lucky player would get their favourite combo done at launch, while the others would have to wait for who knows how long. Not to mention that several core classes originated as hybrids.
Plus, level-dipping covers interesting and fairly important narrative concepts. It's quite fair to allow characters who are mostly single-class but have side training in a secondary class, or characters who stopped advancing in a class and started the path in another.
For making sense with training times, there is nothing you can do really. In many games, times flows on its own for level advancement, and PC level up just as you say "overnight". Training rules have been designed many times, but for a lot of games, they just don't work, they just become a detestable nuisance. The only good approach is not to force training rules on any gaming group, because a lot of them just don't gain any fun from them. Have an optional module if you want, but be warned that even the best training rules often become stale quickly.
I'd rather keep the flexibility of level-based multiclassing, and tell DMs that if they are concerned with non-sensible combinations, a house rule like "maximum N classes per characters" (for example N=3) is very simple to handle and just as simple to ignore (if it's your own house rule, not a standard rule).