Answer

This expression, in its original Talmudic sense, has a completely different meaning. In the Talmud, the question is always how to present a dispute: whether to highlight the side that permits or the side that forbids. So in the Talmud this is only a didactic question, not a halakhic one. See, for example, Beitzah 2b, and also Rashi’s explanation there, and many other places.
Correctly, there are some who use this expression to say that it is preferable to permit. In principle, I agree with you that this is incorrect. A halakhic decisor is not supposed to be lenient, nor is he supposed to be stringent. He is supposed to rule according to what he believes is true in the case at hand.
And perhaps one can give it the following meaning: in a place where there is no reason to forbid, one should permit, and in that sense there is an advantage to leniency. Anything that has not been forbidden is permitted. In other words, to forbid you need a reason, but to permit you do not need a reason.