Simple question, are there any video tutorials available that run-through the basics of the mechanics of AD&D 1e? Or perhaps brief, succinct texts/discussions of said basics? I am finding the pages and pages of text of the manuals to be a bit difficult to conceptualize and I would like to see these rules in action.
The AD&D manuals are awful to learn from in an efficient manner. They are wonderful for long, meandering explorations of the game (probably sitting in an overstuffed leather chair in front of a crackling fire, with a snifter of brandy or somesuch signifiers of studious leisure), but when you're just trying to get up to speed for the first time on how the game works they're an impenetrable thicket of purple prose and inexplicably-placed tangents.
And to be honest, it's written that way because teaching the game was never really its goal, since when it was written everyone learned from someone who already knew how to play anyway. So, still, the best way to learn is from someone else.
Mentorship has gone digital these days though. There are a few excellent resources, distillations of wisdom and rules, that you can use to quickly get an understanding of how to make the game go. Two of the quickest and best in combination are:
The most general, but arguably most useful, is the Quick Primer for Old School Gaming. (Free PDF)
It will not teach you the rules of AD&D, but it will teach you want they are for, the kind of play experience and flow that they most effectively allow you to create. Having an idea of what the point of the rules is in the first place can help immensely in absorbing and contextualising the rules when you do read them.
The second most useful resource is OSRIC, which is an acronym for the unwieldy "Old School Reference and Index Compilation". (Also a free PDF.)
What that title obscures is that it's all the rules for AD&D 1st edition, rewritten to be clear, concise, and utterly bare of any fat, the better to serve as a rules reference. It is not written as a teaching document, so it will not teach you why the rules are the way they are, nor will it explain how to best use them, but it will teach you the rules very quickly. In combination with the Primer though, that's all you need—the Primer explains why and how of the rules, OSRIC explains what they are, and together they are a complete-enough guide to running AD&D.
It's worth noting that OSRIC differs from the actual AD&D rules in a few very minor spots, for legal reasons (e.g., the XP tables are almost, but not quite identical). OSRIC won't lead you astray in any substantial way, but it's helpful to keep in mind for when you do read the original manuals and spot one of these tiny differences. The differences are minor enough that you can smoothly move from the OSRIC version of a rule to the AD&D version of a rule, should you decide to, without any disruption to your game.
(Although I've been gaming since 1st edition and have the original books, I also keep printed copies of both the Primer and OSRIC around. They're that useful.)
Once you've got that far, you're probably prepared to run AD&D just fine. At that point, the manuals themselves are no longer impediments to understanding, but are now detailed essays that you can read at length for deeper insights, or you can dip into every once in a while to glean some new trick, techniques, or understanding of the game.
I know you asked for videos, but for combat I consider the ADDICT reference from the Dragonsfoot site to be excellent. It is extensively footnoted with page numbers..
I did a search and found this video on Youtube. If you look at the related videos on the right you will see a lot of AD&D related video by the same person as well as other folks.