Alright, so the "advantage" of having Splinter be Hamato Yoshi himself rather than the pet rat is commonly parroted without much deeper thought put into it. The argument essentially starts and ends under the assumption that a human being is more likely to remember martial arts than a rat, without seriously considering what the drawbacks of this change really are.
Ignoring the fact that this apparent flaw in having Splinter start out as a rat can easily be explained by him having psychic powers before his mutation and/or have him train after his mutation, which just makes the go to defense of having these two characters be the same one a bit weaker, the idea of making it so that Hamato Yoshi mutates into Splinter has a couple of problems attached it itself:
- The first problem is that it will always require very obvious contrivances as to why he is affected differently from the turtles. In the Fred Wolf cartoon, it is pretty obvious from what we can see that Hamato Yoshi should transform into a turtle based on the established logic, it does really matter if they say he's been close to rats before the incident, he clearly came into contact with the Ooze alongside the turtles. In the 2012 cartoon it isn't quite as egregious but the exact circumstances are still clearly contrived to make it happen, as are they in Rise. There always have to be rat somewhere, there always has to be a reason everyone (including the human being who wouldn't be dumb enough to just touch an unknown substance) is covered in ooze and there always has to be a reason Yoshi does not touch the Turtles despite being so close. This may not seem like such a big deal, but it can easily make the origin far more convoluted than it needs to be.
- The second problem is not an inherent one but relates a certain trend that I have noticed, the iterations that tend to depict Hamato Yoshi as Splinter also tend to be ones in which insist that mutations can be reversed without any problems. I'll spare you the details on how many times these iterations could have cured Splinter, just know there are plenty of opportunities. Given this fact, Splinter obviously doesn't have much reason to remain a rat, as he obviously wouldn't want to stay in the sewers. Of course Splinter has to stay a rat to preserve the status quo and because that is what his toy looks like, which just makes it a bit awkward.
- The third problem is that it requires the use of the "you become what you touched" version of the ooze which is a gateway to "does whatever we want" ooze. As the circumstances regarding the initial mutation requires a loose logic and contrivances to be begin with, the subsequent use it becomes more and more inconsistent in how it works and ultimately ends up being an all purpose plot device. You could stick to the initially established effects but like it or not, it is that version of the rules that eventually lead to cars and ice cream being treated as being subject to mutation.
These reasons lead to further problems down the lines, which goes beyond just Splinter's origin. I might add here that the re-incarnation variant is more or less free of these problems, as he starts out as a rat before the mutation, has no conventional means of returning to the form of Hamato Yoshi and it does not require the stupid version of the ooze (though Bebop and Rocksteady created that problem instead).