I'm pretty poor at reading reindeer minds, I guess they would, though. But when we circle back to humans, I would probably instinctively pay attention to shoulder width and height in a similar fashion when seeing a person from behind, and yet I feel those are more fruitfully described as characteristics with a sex skewed distribution rather that secondary sex characteristics, which to me at least better describes beards and breasts. The conclusion I'll reach in that way will be, for most of the range, tentative, and I instinctively know as much. The conclusion will be fairly firm at the tails, but much less so for typical values of either sex. I'm no Dwayne Johnson, and let's be honest, most of us aren't. If I see a Dwayne Johnson from behind, I would indeed be very surprised to learn that person was born a woman even if I saw him in a dress, but for someone, with my shoulders, I wouldn't be all that surprised even if I saw them in gender-unspecific male-leaning clothes. If I saw such a person in female-signalling attire, I'd read them as a cis woman if anything. Maybe not an excessively feminine one, but "this must be a trans woman/ transvestite/ man going for a woman in carneval" isn't a thought that would likely cross my mind, as it would for Dwayne Johnson. That being said, my shoulders are probably pretty average for a male of my subpopulation. If the male average value for a trait is insufficient to sideline the tentative conclusion triggered by a notoriously transient property such a attire, I fail to see how calling it a "sex characteristic" when talking about individuals rather than population means is anything but misleading. Thus, "do conspecifics instinctively pay attention to a trait under conditions of incomplete information" seems to be a rather insufficient condition for that trait being a "sex characteristic" under what I would consider the most intuitive definition.
Tertiary sex characteristics if you like, or more reasonably "sex correlated" traits rather than "sex linked" traits.
Sex correlated traits are, generally speaking, those that reflect dimorphism and are often assumed to be the result of sexual selection. They are not directly the result of sex-based hormones, nor are they tied to sex chromosomes. They are traits that are independent of the process of sexual differentiation in our development, but which express with different averages by sex and often show considerable overlap in the distributions by sex, and produce a material bimodal distribution when plotted without sex being used to divide the population. By material, I mean that the trough between modes is not near-zero.
Secondary sex characteristics directly result from the process of sex differentiation in our development, either as a direct result of the action of hormones or because they're actually tied to sex chromosomes. Secondary sex characteristics are distinguished from primary sex characteristics in that they are not directly involved in the process of reproduction.
It's not uncommon to be mistaken about a person's sex when looking solely at sex correlated traits - shoulder breadth, height, foot size, hand size, and the like. They're generally good at suggesting sex, but they show a large amount of variation. There are plenty of 6' tall women with broad shoulders and narrow hips out there, and there are plenty of 5'7" men with small feet and small hands out there. These characteristics are a good proxy for sex most of the time, but are not reliable.
It's a lot less common to be mistaken about a person's sex when looking at secondary sex characteristics. Most of the time, if you end up being mistaken, it's because someone has
intentionally attempted to
mimic the opposite sex. Secondary sex characteristics - in the absence of intentional mimicry - is an extremely good indicator of sex, with a very low miss rate.
You might see someone with narrow shoulders and swish from behind and assume that they're female. You might even thing "my, what a caboose that gal has!", but when they turn around and have a beard and adam's apple, and are obviously male it's not like you're going to fall over from shock. Most of the time this would result in "oh, that's a dude! Well, he's got a nice behind" and you go on with your day and don't give it much thought beyond "hey I saw this dude earlier, and at first I thought he was a chick because he had this really nice sway to his backside, boy was I mistaken".
On the other hand... if you see someone with a rolling gait, wide hips, breasts, a facial structure that has a smaller and narrower chin, narrower jaw, rounder forehead, rounder orbital sockets, arched brows, narrow waist compared to hips, no facial hair (and no hint of a 5-oclock shadow), no protruding adam's apple, narrow shoulders relative to their hip width, and femurs that aren't orthogonal to the ground... and they end up having a penis... you're likely to be surprised. Not guaranteed, but likely nonetheless.
Both tertiary and secondary sex characteristics act as signals of sex to potential mates. But at the end of the day, it's primary characteristics that are the deal-breakers in reproduction.
And just in case it needs to be said... talking about the role of reproduction in evolution, and in how different characteristics have evolved and the role that those characteristics play within any given species is not in any way passing judgement on individuals who either cannot or choose not to reproduce. That's an entirely different discussion that has no place in this interaction.