If you can get past Watson being a Chinese woman, with Holmes still alive and unaged today but living in New York, then what’s the problem with Mycroft being fit?
There’s also a bit of Fry (but not Laurie) in that illustration.
Mycroft’s physique change seems to have been handled with some respect to source material; the part that bothers me is the lack of seriousness taken to the mystery side. These are not puzzles that have answers that make sense in retrospect. I referenced “The Sign of Three” and the solution was just nuts and the coincidences beyond the pale of that allowed by suspension of disbelief.
That’s what I meant in my comment above: if that illustration depicts what was considered “corpulent” in those days, then there is definitely a disconnect between now and then. Nowadays that man would be considered chubby or husky, but certainly not overly fat.
The man in the illustration isn’t unusually fat by modern standards, but he’s definitely overweight and I’d guess probably obese.
That’s the thing, though: Mycroft was supposed to be grossly obese. That guy is not, at least not by today’s standards.
And by standards of the time Taft was obese.
I couldn’t elucidate it precisely if given to Skald to scald, but there’s something about heft that changes both tenor and demeanor. Sidney Greenstreet was a wonderful Nero Wolfe because he sounded large and also somehow inert. Stephen Fry is like that as well. You can’t imagine him running. He’s a perfect British archetype, having played both Oscar Wilde and Mycroft. Doesn’t Wilde look like he could grow older and turn into Mycroft, the thing he would most hate? And G. K. Chesterton looks like the mid-stage of that evolution.
They are all exactly what people in those days meant by corpulent, which does not mean grossly obese. Modern obesity would look freakish to Victorians. Nor does Doyle ever say Mycroft is, to my memory. He is “heavily built and massive.” He does have a “gross body” but the word was then used to mean “thick.”
So, yes, certainly Mycroft should look and feel heavy and immobile. He represents the soundness and the weight of the British government. It’s the whole older brother/younger brother division of labor in British society of the time. The older brother inherited the land and the title and the hundreds of years of history tied to them, while the younger brother went out to the colonies and make new history with his wits. Sherlock is the younger brother par excellence; so is Watson, who literally comes back to London from adventures in the colonies and has a wastrel older brother. They’re both symbolically modern; Mycroft is symbolically the past, the best part of it. Everything Doyle does in the Holmes stories is symbolic of his view of England.
Since Watson served as the narrator and POV character, he was rarely described. In fact, I only remember him being described twice. In “Charles Augustus Milverton” he was described as “a middle-sized, strongly built man—square jaw, thick neck, moustache…” and in “His Last Bow” (the chronologically last story, set in 1914) he was described as “a heavily built, elderly man with a gray moustache.”
Few are bigger Sherlockians than I am, but characters’ physical appearances don’t matter much to me. Mycroft not being fat doesn’t bother me any more than Watson not having a moustache, Sherlock not being rail-thin, or Lestrade not resembling a ferret. I care more about the essence of the characters and the stories being brought to the screen in a unique way, and for that I quite enjoy both Elementary and series one of Sherlock. I enjoyed the first Downey Jr movie (have not seen the second) for reasons unrelated to any resemblance it bore to the canon, which was little.
Sherlock’s Mycroft works for me as a modern-day version of the corpulent thinker of the canon, with just enough of the drive that would be required to keep his position in a modern government (Google having been invented since the Victorian era, the canonical Mycroft has lost some of his value). Elementary’s Mycroft works for me as a character who is by nature sedentary and theoretical as in the canon, but who is not satisfied at all with that (an idea brought home at the end of season 2 when he lamented a childhood memory of Sherlock and their father mocking his lack of drive), and who has worked hard to rise above his natural tendencies.
In his first appearance, in A Study in Scarlet, Watson’s friend Stamford describes him as being “as thin as a lathe and as brown as a nut.” That’s when he was just back from Afghanistan and still recovering from his wound, which probably accounts for both the complexion and the thinness. There’s the strong suggestion in Stamford’s words that this isn’t how Watson typically looks.
I’m the same way. In fact, I tend to skip over lengthy physical descriptions of people when I’m reading. It occasionally causes problems. Sometimes an author will switch things up. Rather than using characters names, they may say something like “said the blonde” or “the taller man smiled.” And it always confuses me because I never bother to memorize which girl is blonde or which man is taller. 
An old professor of mine said something once that I heartily agree with: “Unless it’s relevant to the plot, like the hero has a birthmark that proves he’s the true heir to the throne or something, what literary characters look like is completely unimportant.”
There are all sorts of readers. For some, descriptions are a crucial part of the character.
That’s the real issue. Some readers want everything described, some think it’s boring. Some want a and some want b. Then there’s those who want m. No writer can write for all these contradictory needs. I think that’s a huge reason why some people find some writers unreadable. It’s not anything the writer did wrong; it’s that the writer consciously or unconsciously focuses on a different basic type of reader.
Cormac McCarthy is the poster child for this.
The actual quote is “Heavily built and massive, there was a suggestion of uncouth physical inertia in the figure”
I always read this to mean a very tall and big man, who was also overweight, but in a way that emphasized his size, not necessarily in a way that defined his size.
I never imagined him to be say… 5’10" and 450 lbs with no muscle tone, etc… I always imagined him to be tall and large framed, as well as big and fat. Sort of a lot like Stephen Fry was in the Sherlock Holmes movies- maybe a tad fatter, but that’s it.