I still remember Spencer's speech therapist extolling the virtues of separation to me when he cried. She told me that it was good for him to separate from me, and that as his mom I just needed to "be strong" and let him go. He was barely three at the time. It's little wonder that that relationship ended, badly, a few months later.
I accept and in fact welcome the fact that my children are temporarily dependent on me, to varying degrees. This stage is so short and so precious! I am enjoying having them close, and being attached to them. I am likewise enjoying watching them gain their independence in their own time in their own way.
Independence is important, no doubt. But trying to force it? What's the price? It should - and will - come naturally and gradually - if you let it.