Yes, many AA members fly more on CX than AA. In fact, many pay more in order to get mileage.

Banking miles on a FF programme also means the passenger expects a certain FF programme to be more useful to him/her. There is, however, no double-edge sword. Crediting CX flights on AA also means getting less recognition from CX but enjoying more perks on AA. If the passenger buys a CX ticket on CX flights and credit the miles to Asia Miles, CX needs not share the revenue with its partner. CX, of course, values that passenger more.

The same applies for CX's partners.