Nice video, but it makes it hard to review parts of it.
Also, I didn't really get WHY you are proposing the change.
I mean, I agree that random is random and 'card destiny' is kinda stupid, but tradition is also important - at least regarding the premature board cards. I like the idea of fixing a misdeal, when possible.
I'm going to go a long distance to make my points, so bear with me here.
First, consider a game of blackjack, where a novice at third base makes a bad decision which changes the cards the dealer gets, 'causing the players to lose'. Few players will consider the randomness, or that the deck could have been shuffled slightly differently so that the novice's bad play helps the table. Similarly, few players remember the times that the novice's bad play DID help the table. I.E. Your randomness argument is going to be a very hard sell.
The burn is intended to hide the next card making card marking less valuable. Is that really an issue these days? I'd like to think not, at least not in commercial casinos & card rooms, but could be an issue in underground and home games. So consistency is still desirable. But as far as fixing problems goes, burns are just cards.
If there's a misdeal, why can't it simply be fixed? A player is missing a card? Give him the next card. Extra card? Have the dealer randomly select one card to discard - and since you don't know if the player looked at his hand, expose the discard to all players. Multiple problems? OK. Misdeal.
But for the premature board question, I'd like to use the example of a premature turn. You burn another card, preserve the river, then shuffle the premature turn back into the stub. But at that point, there are generally two schools of thought: Use the river as the turn, or deal a new turn using the preserved river as the intended river. And now a third option is being introduced, it's random either way, so shuffle the premature card as well as the unused river and burn.
The option of preserving the river and using it as the river as 'destined' is the worst option because it will be sitting there without a burn to cover it, it might become fouled/mucked, and it provides information - that the original exposed turn, having not become the new shuffled turn, isn't going to be the river either.
Using the river as the turn provides less information - the value of one card that may or may not become the new river.
The suggested new random method of reshuffling all unused cards before the new turn provides the slightly more information - the value of one card that may or may not become the turn OR river.
A premature flop has all the same problems and solutions, just more cards.
All three solutions have their problems and merits, so why go against tradition?
Personally, I prefer to stick with the option of using the river as the turn (or the turn/burn/river to replace the premature flop). It's what players and dealers are used to, and it's the best solution for a reason generally not discussed here: It's the fastest method. I.E. You can replace the turn or flop, then begin shuffling while the action on the replacement is taking place so the newly shuffled stub is ready when needed.
Bottom line, some rules are all about Hands Per Hour. And that makes management AND players happy. And it's the primary reason I like the idea of changing the misdeal rule. Particularly when you consider that most misdeals happen before the shuffler is done, requiring a time consuming hand shuffle.
On a side note, have you ever dealt heads-up and dealt to the button first? Should be a misdeal, but knowing neither player looked, you simply switched the hands. Is that acceptable?