I agree that it would neutralize body issues, for most people at least. I'm not sure about the second paragraph. Those feelings will still be there, they're not felt by people of the same gender because supposedly most people aren't attracted to the same gender. Evidence to this is the discomfort felt by those who actually are attracted to their peers. It makes things awkward for them.
I'd support it if I didn't abhor the idea of changing in front of other people.
I'm not sure exactly what you are referring to by "those feelings"?
I think I can give you a response about the rest of your sentiment: I believe that 'discomfort' felt because of co-mingling with the politics of attraction involved, is a part of our culture that desperately needs mediating and more extensive development, and although I have not identified the specific workings of how such a maturity would develop in a culture where this was experiencialy shared by its members, I do think this type of experience could find the answers to a lot of issues around:
values that produce successful relationships between peers.
values surrounding the issues of self-validation in the context of sexual-competition (which when you try to sideline and whitewash from a culture, produces the worst kind of silent-establishment of a value-laden-hierarchy: that can't be overcome by any means of engaged address to that underlying situation: which can only accessed from offering direct challenge to those [imprinted] constructs of social-value).
To offer some kind of illustration of both of these topics: I would point to phenomena where peer-based relationships offer the best models of interdependence, as a framework conducive for generating homogeneous values that don't veer into mutually destructive divergences: but these sorts of relationships don't fit into the most venerated or common relationship tropes- if anything, they are too mundane to foster or cultivate because of the pressure offered by more alluring relationship dynamics which are inherently scarce (and usually have some degree of costliness or riskiness- even if that cost or risk is secondary: the fear of losing access to some scarce [and valuable] quality). Largely the value attached to scarcity is manufactured by its projected qualities when those qualities are kept hidden:
And now we seem to be at the crossroads of quite a dilemma-
Can culture support a masculine identity that can fully weather social and biological factors 'he' could attend to and must contend with, to avoid establishing a patriarchal edifice of security- bought from the arbitrary limitations put onto cultural themes of 'conventional' co-ordination ('arbitrary' when surveyed within the feminine treatment of the same social factors)(post script: the youtube video I refer to later cites an "in group", such a construct is the basis for discarding a measure of potential in expressible liberty, to access a measure of some appearance of security (which only offers a facade to use through a default conventionality, that must either be diffused by intoxicating substances, or some form of ambiance conducive to a self-imposed-extradition, in order to forge genuine sentiments not dressed-up by the facade's judgements on fashionability:- which displaces an articulation of authentic sensibility (and dislocates people from an internal reliance upon an immediate sense of engaging your own intelligence to the consideration of perceptions)).
To reduce this into some semblance of relevance to what you originally put to me: (and I know this only illustrates partially what I'm trying to discuss here)- Is the boy anxious because the girl he likes will get a preview of (a vaster sense of) his social-value, before he has managed to design some way of surmounting the vulnerability that presents itself as an (comparatively) unfavorable social-value. Social-value is constructed from operative themes (or patterns) noticed as a latent dimension to behaviors, and it can mean a great deal for an individual to endure a social-value which is based in erroneous thinking; inversely, the unfairness of a comfort visited upon a select, is not grounds for self-pity or vainglory: and the indulgence of either is a short-lived 'cult of decadence': which can only be prolonged by dredging further in the illusory space created by keeping the consideration of qualities hidden, from confidently making assessments- for a full knowledge on the matters involved (which also include internal cognaziances with regard to the external matter of some social-mechanic).
On these grounds, woman do have an inherently easier time- I might aswell just post this youtube video because it contains some of those factors, which I would underline with a different contextual emphasis (while ignoring some the moral narrative in this youtube videos's setup, also because his narrative doesn't include the focus on McLuhan factors on the developments in the progression of cultural evolution):
let me see if I can condense this argument into multimedia:
Sociologically speaking, economic 'reality' is more important from the masculine identity perspective- because:

-when finally resolved;- would mean reducing the field upon which a man could make himself indispensable to a woman; because biologically, men have a proven history of being far more proportionately dispensed-with; whereas it has been proven that woman have been treated to a far greater rate of
indispensability (60% of all men descended from our human ancestors, did NOT father children, as opposed to 20% of woman who did NOT mother any children). And I'll just leave this youtube video here for some further facts on biological factors (I'm not sure about all the conclusions that are cited in these videos on sociological matters, but the biological information is quite interesting).
https://www.youtube.com/watch?v=iwzvqL0TxEQ
My further discussion on my last point with the treatments of indispensability:
Because of how systems of power and the pattern to which power can be correlated with, have a larger potential imprint factor on the mind, as an environmental temptation: its easier to fall victim to the narrative of how men- must try to distinguish themselves fervently in order to "win" affections and attention, failing to distinguish himself on the grounds of doing something that offers a proof of mastery, or taking great risks for novel rewards, brings on pressured feelings due to the looming threat of being dispensed-with, in fulfillment of some objective sense : that has supposedly been deselected in all his inherited and 'collective' experiences of masculinity (by the culture he inherits, with it's language that endures with all its social-DNA inbuilt within it) toward making some distinction between him and the other 60% of the 'tribesmen'. Are the kinds of conditions that can only be remedied by getting to the point where culture is malleable to the individual, through the conduit of homogeneous values that can only ever open themselves to adaption after they are taken entirely for how they present themselves, which can only occur with a bedrock of truth- to hopefully produce the edifice for moving onto having itself adapted liberally as knowledge is expanded to create a further potential whereto express liberty.
When reality is mundane, novelty is to decide a chosen adaption with reality. When reality contains the novelty, decisions are mundane unless they are choosing to slavishly accord with the novelty which "reality" is then containing the source of (by your style of chosen acquiescence to the "reality"). So therefore;- help the world become more mundane. This is my answer to SHOULD unisex locker/bathrooms be a thing. Obviously pragmatic considerations of HOW WOULD it be a real thing, is a vastly more specialized kettle of fish per the local offering of implementation.