We need to be concerned with both for many reasons than I can list here.
Total Testosterone is important as we want to see how much in aggregate the body is producing in serum. This is especially important as it relates to transport hormones like SHBG, Albumin and Estrogen conversion.
Free Testosterone is critical in that it tells us how much of the Total Testosterone is available for the body to use. In other words, the stuff that makes the magic!
Doctor's who don't test for Free T are the ones who simply are not trained correctly unless they are calculating Free T from other labs (and this I doubt most GPs or Endos know how to do this correctly).
Just look at how many GPs and Endos use TSH only to evaluate Thyroid health...and for those that are trained correctly know that TSH is a very very poor marker for evaluating Thyroid health.