I don’t want to be all Debbie Downer, but I think you have a big choice to make here: either raise your budget considerably and get some kind of pre-packaged kit that is close to a “turn key” solution, or buckle down for a long and time consuming DIY project (which is probably still going to cost more than $300 unless you get a lot of stuff for free).
I’m only bringing this up because the impression I got from your post was that you plan to spend a few nice Saturday afternoons out in the garage “in your spare time” to build an arcade machine, and it gets done when it gets done. While you can build shelving units for your basement in a Saturday afternoon or two, be warned, MAME cabinet projects are absolute time vampires by comparison. Online researching, hunting down parts, ordering parts, designing the project, making all the big cuts, assembling the project, designing the artwork, 4000 trips back and forth to Home Depot, acquiring all the tools you realize you need but don’t already have, running into unforeseen problems, solving them, etc. etc.
My point here is not to make the undertaking seem like some kind of hellish ordeal, because it’s not at all, so long as your heart’s really in it. In other words, it’s a lot of work--especially for a first time builder—and it has to be a labor of love.
I’m not really familiar with the XX-in-1 boards, but from the sounds of what has been said already, you may need to use MAME if you want to save high scores on a large variety of games (I do it on my cabinet). MAME is superior in just about every way to the XX-in-1 boards anyway, so it’s a good route to go if you are willing to spend the time fiddling with the software to set it all up (kind of intimidating at first, but really not that bad at all once you get into it).