My thinking was pretty much the same as yours. The whole movie, both sections, has a very dark and haunted "feel" to it. There's a moroseness to the Istanbul part which mixes well with the darkness of Transylvania. I don't understand why the second part is considered strange honestly. If we consider Young Indy to be the youthful adventures of the guy in Raiders through KoTCS, why shouldn't he have an encounter with the supernatural? I think a vampire is no less believable than the Sankara Stones or Aliens. Also, I enjoyed the Transylvania section for a few reasons: I love the group dynamic, it almost feels like watching an RPG in a sense in that you have this party of highly skilled characters; I thought Maria made for an interesting woman, not the demure female Indy is used to at this stage (compare Maria and Molly). I love the fact that Dracula is played much differently than he is usually, especially for the 1990s. This came out around the same era as the 1992 Batman, showing Vlad as a seductive Prince, Interview with a Vampire which showed these hunky guys as very sensual vampires; it was refreshing to see a different approach; the rat-like and decadent sociopath feel of this vampire made him creepy. It also felt like a horror B-movie and reminded me of the haunted castle segment they had wanted to do.
Lastly, I feel Flannery was written much like Harrison's Indy throughout the movie, especially in the Instanbul part. He has the smugness, arrogance and self assuredness of his older self; in the linking segment he has the older Indy'sdark cynicism: "Yeah, well, at least you got some sleep." "It's not like I have a choice." His attitude toward the bureaucrats in intelligence, which gets him transferred, also feels a lot like how a young Indy would, shades of "bureaucratic fools." Indy is at his best when he is a darker character and not a wide eyed sweet kid.
I'm one of those people who went back and forth on Young Indy for years but I've come to love it and hope Disney keeps it canon. I actually don't think Indy meeting all these historical figures is that outlandish. Is it really any more outlandish than the same man discovering two of the biggest relics in Judeo-Christian religion only two years apart? Is it really that outlandish compared to the films, where Indy gets his fear of snakes, his scar, and his hat, and the blue print for his adventure gear all in the space of a single afternoon? Indy lives in a fantastic version of our world, where the supernatural is present and apparently, there are many forces at play - From aliens to various Gods (TOD showing us that the Hindu gods are likely real). This is a man who can survive being dragged by a truck (while nursing a bullet wound), survive slamming into a stone wall after probably going through the rapid fall of a bridge, and who survives a nuclear blast. Meeting people who would later become famous is no more outlandish than any of that.