There is an enormous amount of noise being made about the alleged advantage that transgender women have in athletics. First, let me start off by saying that the topic of athletic performance is not an area of expertise for me, so what follows is largely my opinion, bolstered by observations about what I see in the academic materials published.
Broadly speaking, the claims of "an advantage" hinge around the development of secondary sex characteristics from a male puberty. Generally speaking, these claims fall into a small range of categories relating to skeletal structure, muscle mass, and height, and so on.
A superficial look would suggest that a male certainly does have "an advantage" in athletics, and to some extent that is borne out by marked differences between male and female record performances in various forms of athletics. However, it's a much more complicated discussion when we start talking about transgender athletes.
Is There An Advantage?
First, athletic performance in general is clearly a multi-factor issue. Everything from genetics to coaching and mental health appears in a search for 'athlete performance' on Google Scholar. Clearly, even the notion of athletic performance is non-trivial, and we certainly should bear this in mind when talking about the question of whether or not an athlete has an "unfair advantage", regardless of their gender status.
In other words, we can't simply make the claim that someone who once had masculine levels of testosterone has an "intrinsic advantage" in all forms of athletics. That simply doesn't make sense, since there are a plethora of factors involved in athletic performance, and we don't even have a workable model for understanding the degree of effect each element might have on an individual athlete.
When we are looking at transgender athletes, there are even more topics to consider: when did the person begin gender-affirming care such as either puberty blockers (for younger transgender people) and hormone replacement therapy (HRT)? For example, someone like Jenna Talackova who began transition early, has a visibly different skeletal structure to a transgender woman who started their transition later in life. How that influences her athletic capabilities is an important question worthy of exploration.
Specific Alleged Advantages
On the matter of muscle mass, feeding queries to Google Scholar about the effects of estrogen on transgender women and muscle mass, we find all sorts of interesting papers, which largely come to "well, basically transgender women don't have an advantage in terms of muscle mass / strength.
Now, let's do a basic little bit of analysis here. Let's assume a female transgender athlete has a generally masculine skeleton (so, somewhat heavier, with longer limbs than is average for female of the same height). That same athlete has been on HRT long enough to qualify to compete in a domain like swimming. We know from various studies that her muscle strength will be comparable to that of a natal woman, and she is pulling the heavier density skeleton through the water. Further, there will be some hydrodynamic differences resulting from slightly wider shoulders, and longer arms. Does the "cost" of the disadvantages (hydrodynamic penalties, extra skeletal weight, and reduce muscle power) get offset by the mechanical advantages of slightly longer limbs, and larger hands / feet?
If your answer was "yes" or "no", you're probably over-simplifying the situation in your mind. Yes, the longer limbs will provide _some_ mechanical advantages, but it's far from clear that they would entirely offset the penalties. The only "correct" answer here is "we don't know" because frankly, nobody has bothered to measure it yet.
The Hormone "Advantage"
Much of the hue and cry going on here revolves around the idea that testosterone is some kind of magic elixir when it comes to athletic performance. There is an additional dimension which boils down to a rather odd bit of reasoning that "if you ever had testosterone in your body, you have an unreasonable advantage". That simply doesn't make sense. Again, just as with athletic performance as a whole, there are a series of factors at play here that shouldn't be ignored or simply trivialized away.
First of all, the effects of HRT on the body vary wildly across individuals, and it varies even more based on the age that one starts taking them. Someone who starts taking them early in life will ultimately have more of the characteristics of their target sex than someone who starts taking them later in life.
Then we get into the whole business of what happens after genital surgery for transgender women. Ironically, a transgender woman will actually have significantly lower testosterone levels than a natal woman has because they have no gonads at all, and just as testicles produce a small amount of estrogen, ovaries produce a small (but measurable) amount of testosterone. So much of the discourse around this matter seems to presuppose that the transgender woman's body is forever filled with testosterone and this is simply not true. (and even pre-operatively, testosterone blockers are used to medically suppress testosterone production down to 'female normal' levels.
That "advantage" simply doesn't exist after a certain point, and it's unreasonable to treat it as an advantage for rhetorical purposes.
Returning To Athletics
The article from The Guardian linked to above makes the rather dismissive claim that the reason that transgender women haven't been dominating in athletic competition is because "the athletes to date simply haven't been all that good". Considering that we have had transgender athletes competing in high level athletics (as in pre-Olympic level competition) since the mid-2000s, that seems to be just a tiny bit ridiculous. You don't get to that level of athletics by "not being that good".
At this stage, we simply don't have enough evidence to even make the claim that transgender women have any kind of advantage in athletics, much less defining what that advantage might actually look like in any particular sport. We have a lot of claims that there is an advantage, yet if such an advantage were a thing, then one would expect that transgender women would generally be outperforming their peers in competition.
Again, a glance through the literature in Google Scholar simply doesn't turn up any papers that show that to be the case. The relative lack of transgender women on the podiums suggests strongly that they are performing within the bell curve of athletes at their level, not significantly above it as those who advocate that there is an advantage would have us believe.
This is clearly not an easy task for scholars to sort out. Not only must the factors that influence athletic ability and performance be much better defined, but then on a sport by sport basis, it is essential to better understand how those attributes affect athlete performance in those sports. You need a very solid basis upon which to do comparative analysis here before you can actually say that a transgender woman actually has an advantage that falls outside of normal variation.
Just claiming that there is an advantage is not good enough here. We have evidence that there are mitigating factors that offset supposed advantages, and may do so to such a degree as to cancel any apparent advantage out, or minimizing it to the point where it falls well within normal variance for athletic performance. If you want to claim an advantage exists, not only is it essential that you be able to show that the advantage exists, but also that it grossly exceeds normal variances within a given sport.