A chicken and pheasant hybrid is not going to live a long and healthy life. Inbreeding can be a problem if you're not keeping records or banding birds but it's alot easier to fix than bloodlines being genetically polluted.
Crossing chickens with pheasants gives you sterile mules. If any eggs hatch the young will be weak and likely will die for the most part. Even the ones that make it through the first year don't live very long. Hybrids are damaging to the pheasant hobby as a whole.