The reason is: the impedance at the solder point is not 300 ohms and
with your balun, you're inserting more losses .

True, but without a balun, one is introducing common mode effects.
I, personally, would use a 1:1 choke at the balanced to unbalanced
connection point.

There sure is RFI coming from the coax . I should try it. I'll
disconnect my coax and try 10 turns of coax as a balun. I'll know for
sure. Or would the "bead baluns" be better ?