I had the same thought when I spotted a 100-300mm at auction - a possible good economy alternative to the DA 300mm.
Check which version of the sigma 100-300mm it is. I have, I think, the first version, pre digital. It pretty much matches my DFA 150-450mm 100-200mm, but at 300mm f4 iq is a little off (see
this post), killing it for tc use. At 300mm f8 it is tack sharp again. I did actually send an online inquiry to sigma re a check/collimation - but they won't do discontinued lenses and it's not clear that the cost will result in any improvement..
I think the later DG versions are better. If your use is primarily going to be at 300mm then that lowers the utility of the zoom...
Oh and btw, birding at distances 20m and more, really what you want is 600mm!