The problem with feat chains as a way to express a "prestige class" or other concept is that, in that capacity, they lose their value as differentiators and customization. I.e. if you are spending all your feats to express that you are an Order of the Bow Initiate then you don't have any left to be unique yourself.
That's the problem that my halfling rogue/sorceror player would have with this system. Maybe he can take 2-3 feats to get his spell "talents" (and those would be very powerful feats if you think about it) but then those are all of his feats for the first 6 levels of the character. Part of his character concept is that he dual wield daggers (his "claws") and he doesn't have any feats left to express that with. In the end, trading BAB, sneak attack progression, etc. for a couple spell abilities makes a lot more sense both gamewise and RP-wise for his character.
A possible fix, as was mentioned, is simply to give lots and lots of feats. Actually I think such a system would be fine. However, at that point, you have just taken your original problem (if I allow lots of multi-classing then players might abuse it by dabbling in too many classes) and made it worse. Now you have an even more open-ended system than the one that was originally being complained about and your characters may have even more opportunities to "game" the system.
In the end, the multi-classing system itself works pretty well as long as you make sure your players don't let it get out of hand (by, for example, pre-approving multi-class choices). It gives a little bit more structure than this an open-ended, massive list of feats would and yet it also provides some more flexibility and trade offs (i.e. you can vary skill points, hit die, BAB, etc. with classes and you can't with feat chains). If it ain't broke, I'm not sure why you'd want to fix it.
That's the problem that my halfling rogue/sorceror player would have with this system. Maybe he can take 2-3 feats to get his spell "talents" (and those would be very powerful feats if you think about it) but then those are all of his feats for the first 6 levels of the character. Part of his character concept is that he dual wield daggers (his "claws") and he doesn't have any feats left to express that with. In the end, trading BAB, sneak attack progression, etc. for a couple spell abilities makes a lot more sense both gamewise and RP-wise for his character.
A possible fix, as was mentioned, is simply to give lots and lots of feats. Actually I think such a system would be fine. However, at that point, you have just taken your original problem (if I allow lots of multi-classing then players might abuse it by dabbling in too many classes) and made it worse. Now you have an even more open-ended system than the one that was originally being complained about and your characters may have even more opportunities to "game" the system.
In the end, the multi-classing system itself works pretty well as long as you make sure your players don't let it get out of hand (by, for example, pre-approving multi-class choices). It gives a little bit more structure than this an open-ended, massive list of feats would and yet it also provides some more flexibility and trade offs (i.e. you can vary skill points, hit die, BAB, etc. with classes and you can't with feat chains). If it ain't broke, I'm not sure why you'd want to fix it.