I'm afraid the conflict is more direct that you're making it out to be.

In most cases, we use language not with precisely drawn boundaries, but as analogies to a canonical example or an accepted set of variants--kind of a Platonic ideal of what the word means or what the typical range is.

We also form associations liberally while speaking.

Because of this, inclusive language doesn't necessarily come for free.

One way it doesn't necessarily come for free is if you have to use dehumanizing terms where otherwise you could use empathic terms. For instance, if you think your local schools do a dreadful job of accommodating a girl's first period...I mean...accommodating a pre-menstrual individual's first menstruation.... Well goodness, isn't that awkward language! It threatens to strip all the delicacy and sensitivity out of the matter.

Another way it doesn't necessarily come for free is the hazard of diluting key issues. For instance, if we're talking about women's health and we can immediately and without caveats think about cervical cancer screening, it is different than if we have to parse things apart into, okay, women with a cervix, and women with a prostate gland, and.... It raises the danger that a critical issue will fall on the floor because the division suggests that equal attention should be paid to equal problems...except the numbers are wildly skewed in one direction.

A third way it may not come for free is that it is at least plausible that it can provoke a (presumably weaker) form of gender dysphoria in people whose identities have gotten broadened to include a self-image that they do not identify with. For instance, if a cis man has no convenient linguistic way to refer to himself as the 100% biologically and cognitively typical man, are we actually completely sure that this is not going to cause him distress?

Now, this isn't to say that these issues should drive our response. But what we shouldn't do is assume that they don't even exist. If people express concerns, they should be taken seriously at first. Very often people have concerns that are invalid (c.f. anti-vaxxers). Our job then is to reassure them. (But note that you cannot reassure an anti-vaxxer by telling them that they are wrong about their fear of being vaccinated. The fear is unfounded, and we may need to insist that they face their fear or whatever, but if they're actually worried we can't simply declare that they're not.)

So I think the "no conflict" stance is overstated. Obviously there have been a lot of cases where more conflict has been infused into the situation than is warranted. But I think it is nonetheless important to listen to people's concerns and keep in mind that they may have some degree of validity.