It has always been put forth that the killer had some kind of medical training as the organs were removed with the skill of a surgeon.

Hmm... it fits. I mean look at his beard! Not taking the mick but only a nutter would grow a beard such as that.

'Dr.' Tumblety definitely was not overlooked as a suspect. He was considered such a prime suspect that when he fled England, Scotland Yard sent two members to NYC to keep an eye on him. This they did, with help from the NYC Police Department until Tumblety slipped away and disappeared from sight for a while. This is made clear in a book I have in my possession that contains the remaining documents from the Scotland Yard File on the case.
Scotland Yard had, for the time, done as good a job as was possible in trying to ascertain the Rippers identity, even going so far as to check out some Portugese Butchers who had been to London while working on a ship that was bringing imported cows from the European Continent.
This does not mean that they didn't make mistakes: they did.
But you have to remember that they didn't have the forensics then that they do now to help identify killers or the people to study the psychology of people who committed regular crimes, much less someone who committed the heinous crime like Jack The Rippers. Even the tactics of questioning suspects was not as developed then as it is now, though some individuals who were part of Scotland Yard at the time were very adept at it.
Another question regarding Jack the Ripper is the total number of victims he was credited with. The number most people agree on is five victims, though it has ranged from as few as three to as many as fifteen. And even the total length of time he was out committing the crimes he was accused of varies by what source you consult. Some say he had started out in late 1886, while others say his last victims weren't killed until either 1891 or 1892. Regardless of the total number, the barbarity of his crimes have rarely been equalled.