I am not a lawyer or anybody qualified to give you an actual answer, but my tentative opinion is that this is not really permissible because of the method you specified for getting the game.
Since you did not specify a jurisdiction I am going to assume U.S. copyright law applies, because it is the most populous nation of native English speakers. You might also want to keep in mind that you would be likely to be tried in the 9th Circuit, since that's where most of the big copyright cases happen, due to most copyright holders being headquartered in that jurisdiction (Redwood Washington, Hollywood California, and the Silicon Valley come to mind)
Downloading a R.O.M. is a flat out violation of U.S.C. Title 17 §106 because it is necessary for you to reproduce the file by copying it onto your computer in order to be able download it. There are some exceptions to this rule such as fair use, but they constitute an affirmative defense since you have to admit to the principle fact that you created a reproduction in the first place.
We do not even get into the fact that you do not plan to redistribute or alter the files, because it is illegal for you to obtain them in this way in the first place.
Subject to sections 107 through 122, the owner of copyright under this title has the exclusive rights to do and to authorize any of the following:
(1) to reproduce the copyrighted work in copies or phonorecords;
Note that a popularly cited series of cases as it pertains to computer software is the Mai Trio. This series of cases includes Mai Systems Corp. v. Peak Computer, Inc., Wall Data Inc. vs. Los Angeles County's Sherriff Office and Triad v. Southeastern Express. To summarize the significance of these cases, authorized copies come with an implied license to make temporary copies as is necessary to use them by the licensee, but unauthorized copies do not and you may not exceed the extent of authorized use. The very means of obtaining these copies is illegal, and playing them is also illegal.
You do not have much in the way of a fair use argument that would exonerate you. Your only interest in the games may be for a nonprofit educational commentary, but those the only factors you have going in your favor, and the entire character of use needs to be considered so that may not be enough of an excuse:
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the
The copyrighted work is a commercial game, and you are using it as a game so your work is not transformative. You need to download the entire game in order to play it. The effect upon the potential market in downloading these R.O.M. files is that you do not have to pay anybody for authorized copies. Moreover, there is relatively little about the purpose and character of your use which requires you to download the copies from a third party source: It is just convenient for you. This is not quite the same as taking a screenshot of a game and showing it off to prove a point to others. The character of use is so that you can personally play the games.
Maybe an argument could be made if you copied your own games. U.S.C. §117 has express provisions for making a backup copy of a game you own, although they were ruled to be irrelevant in Atari v. J.S. & A., Inc., which regarded cartridge to cartridge copying effectively illegal. Nintendo also managed to sue the pants off of the creators of the Game Doctor, which copied from cartridge to floppy.
Hypothetically, able to argue a case analogous to R.I.A.A. v. Diamond Multimedia that a mere cartridge reader to interface with a computer which does the copying is not a dedicated copier, and that copying your own copies for the purpose of format shifting constitutes fair use so long as you do not exceed license. However there is also a good chance that these could be ruled illegal too, and I do not know of a case which directly tests its applicability to games.
How do Youtubers get away with it? Well, copyright is civil law, so it is left up to the copyright holder to decide whether or not to sue, and it often is not worth the hassle to go after small time infringers. Publishers also might not know for a fact that they downloaded the games.
However, just to simplify matters, if you must use an emulator, what you should probably do is go a generation up and use a PlayStation emulator with the original game discs. That is expressly legal according to existing precedent such as Sony v. Connectix and Sony v. Bleem. Now granted, both of those companies ended up going out of business after the lawsuits, but that was largely because of extralegal reasons and the precedence set by these cases should make a future case easier to win if push came to shove.