In my limited experience (DNA extraction from cultured cells) the single biggest factor in efficiency of any DNA extraction protocol is the extent of cell lysis. Even with pelleted cultured cells where there's no collagen network the efficiency can be severely affected merely by how well you resuspend your cells before the addition of lysis buffer. I do not know for sure but I can imagine red meat has a more robust Collagen network than chicken so the initial task of disrupting the tissue (to allow for effecient lysis) will be more difficult with red meat. If you are using a column based isolation method the amount of un- or partially lysed material will severely hamper both the quality and quantity of DNA extracted. Perhaps as a first step, cut your sampes into small fragments and treat with collagenase whilst stirring?! Just an idea.......