Hmm, I can see how you got there, but I don't think it has worked out that way in my case.
The amount of time you have to spend writing grants is a function of: 1) how much money you are burning, 2) how much help do you have writing, and 3) how good is your science (better science will be more successful).
No doubt I hurt on #1. But I think I help enough on #2 and #3 to compensate, and therefore I don't think my PI spends a net larger percentage of time writing with me than he would without me. Anecdotally, over my time with him, I think has been a slow decrease in the fraction of his time he spends writing.