Perhaps you have seen this riddle.
A father and his son were driving to a ball game when their car stalled on the railroad tracks. In the distance a train whistle blew a warning. Frantically, the father tried to start the engine, but in his panic, he couldn't turn the key, and the car was hit by the onrushing train. An ambulance sped to the scene and picked them up. On the way to the hospital, the father died. The son was still alive but his condition was very serious, and he needed immediate surgery. The moment they arrived at the hospital, he was wheeled into an emergency operating room, and the surgeon came in, expecting a routine case. However, on seeing the boy, the surgeon blanched and and muttered, "I can't operate on this boy--he's my son."
How can this be true? The answer is that the surgeon is the mother of the injured boy, thus pointing to the assumption that many have that surgeons are males.
Arguments about the discontinuance of the gendered suffixes, such as -ette or -ress, and the prefix adjective "woman," as in "woman surgeon," is understandable. I have noticed that a female actor is now called just "actor," instead of "actress," and "drum major" instead of "drum majorette." The suffix -ette, of course, means "small," so is unacceptable. Still, though, it is quite acceptable and usual to use "headmistress" and "policewoman" and "firewoman," which I suppose some prefer, although "firefighter" is coming in use. But, still, "chairman" instead of "chair" seems to persist. But, ships now are not usually referred to as "she," which I always thought unacceptable and hopelessly old-fashioned.
I suppose the elimination of sex descriptors continues the inclusive trend that began centuries ago, perhaps in Middle English, when nouns had gender, as is still true today in Spanish, French, and Russian. I have experienced learning Chinese and know that it does not even have gendered pronouns--no equivalent of he, him, she, her.
We have the curious practice occurring in written English concerning the use of gendered pronouns: changing the number of the pronoun to plural even when it refers to an antecedent singular noun if the gender is unimportant; e.g., "The student was late when they had to wait for the train." And, in academic writing there is a counter-trend (and overcompensation?) to use "she" and the other female pronouns, for the common pronoun.
There is a version of the New Testament written in neutral language. But, in the film The Matrix, Neo is surprised to learn that Trinity, the superhacker, is a woman. Cognitive dissonance about the innate ability of a woman and the reversal of role expectations is a trope in popular culture.
To the question: Does the gender of a professional really matter?--I'd say not always, or even usually, but sometimes, in myriad ways, surely it does, when the sex itself of the person is involved. As an instance: Last week a female student (here we do not have a noun specifically for them) was late to class due to the time it took for her to change a tampon. She was not able to explain to me her situation (most female students are readily able to) and I issued her a Detention slip. She then must go (missing some class time) to a Counselor whose email I do not see until the next day, and texts her mom who telephones me during class (I know the mom and student quite well). Not sure the relevance of this example, except to make the point (maybe a side issue, maybe not) that gender surely matters in some instances. True?
Still, though, we have the welcome trend to deemphasize gender, which I hope reflects a greater inclusivity in society generally. Yet, when we consider income differentials and the glass ceiling in the corporate world and the number of women in positions of power in government--we know that American society lags behind some others.