Which value is more important to doctors when considering putting patients on trt, free testosterone or total testosterone ?

Hopefully neither: The diagnosis of hypogonadism should not be made based on a lab test alone. Someone in your age group is very unlikely to have true hypogonadism unless they have had a major disorder affecting the testicles or pituitary gland. Free or bioavailable testosterone levels done in an endocrine lab (not a local lab) give the most reliable results. A low normal testosterone level is normal for many people.