Burden of proof in the sense used here is mythical, however. The basic idea here is that 'claim-making' gives an automatic obligation to give supporting arguments. This is not, in fact, something that anyone accepts as a matter of practice, and it is noticeable that in the above video the narrator makes a number of claims for which no supporting evidence is given. Indeed, it is logically impossible for a human being to fulfill an obligation of this sort because every supporting argument multiplies claims. The position that claim-making automatically gives an obligation to give supporting arguments for the claim implies that anyone who makes a claim has an obligation to make infinitely many claims. Since this is impossible, any supposed obligation with such an implication is obviously an absurd candidate for being an obligation. (Infinity could be prevented by allowing self-evident or properly basic claims that require no other claims as support. This would imply one of two things. Either there would be claims requiring no support, and thus bearing no burden of proof, or there would be claims whose burden of proof is met by something other than an argument -- direct experience, for instance. Either route would require a very different account of burden of proof than is given in the above video. And each one creates problems of its own, which I won't go into here.)
Claim-making, therefore, does not create any burden of proof; the burden of proof is not magically created when we assert things. Nor has it ever been plausible that mere claim-making generates obligations in this way. Further, when we look at rational interactions, we find that the whole "burden of proof is automatically on anyone who makes an assertion" falls apart completely. We get the problem that's never properly addressed in the video above that expressions of denial, doubt, skepticism, and criticism all admit of assertion. To handle this, people often fudge by talking about existence and nonexistence. This gives us the account of burden of proof (found here, for instance) in which claims that something exists automatically have the burden of proof. Virtually nobody ever bothers to prove that the burden of proof exists in such cases, even though this is clearly an existence claim and therefore would have the burden of proof. And when people do give arguments they are nonsense like the following (from the same source):
Well think whether or not it is a better way to proceed through life to accept anything and everything that people claim to be so. Experience should instruct every thinking human that there is a high probability that not everything that people claim to be true is actually true. Some claims might be made with the claimant aware that the claim is not true and some claims might be made with the claimant thinking that they are true but being mistaken. As it is for most humans not a very good idea to proceed through life based on beliefs that are false and thinking things to be true when they are not, most humans and those who would use reason to guide them will want some evidence and reasoning to support a claim being asserted to be true. So the burden is on those who make claims to offer reason and evidence in support of those claims.
It can clearly be seen that (1) this would apply equally to claims about nonexistence, or, indeed, any claims at all, and thus requires the more general position that claim-making on its own generates a burden of proof, and carry all the corresponding problems; and (2) this makes the absurd assumption that burden of proof is the only way to prevent oneself from "accepting anything and everything that people claim to be so" when it is in fact obvious that you could just judge in terms of your own assessment of the evidence instead of always demanding that other people do your rational work for you.
In general, the assumption that there is a burden of proof, and that it favors one side, is usually smuggled into the argument. This is often quite clear with more sophisticated arguments like 'epistemic modesty' arguments (of the sort discussed here). OK, so suppose a claim is more 'epistemically immodest' (or whatever you want to substitute for it). It's still an open question whether it imposes any obligation; we need an account linking the two, and this is generally lacking. All the most plausible attempts to give a good account of this sort of general and philosophical burden of proof essentially make it a matter of presumptive reasoning of some kind; but the same problem arises here: your presumptions don't obligate anyone else. Indeed, this could be the slogan for any kind of counter-response to accounts like the above accounts: they are attempts to give high ground to preferred positions by a hocus-pocus conjuration of nonexistent obligations. Moreover, people who propose such accounts are in constant performative contradiction with their own account: they never give their own account of burden of proof the kind of support that would be required if their account were true.
When we look at plausibly rational appeals to burden of proof in real life, we find a mess, and the better accounts recognize this. We get this sort of point from Nizkor:
In many situations, one side has the burden of proof resting on it. This side is obligated to provide evidence for its position. The claim of the other side, the one that does not bear the burden of proof, is assumed to be true unless proven otherwise. The difficulty in such cases is determining which side, if any, the burden of proof rests on. In many cases, settling this issue can be a matter of significant debate. In some cases the burden of proof is set by the situation. For example, in American law a person is assumed to be innocent until proven guilty (hence the burden of proof is on the prosecution). As another example, in debate the burden of proof is placed on the affirmative team. As a final example, in most cases the burden of proof rests on those who claim something exists (such as Bigfoot, psychic powers, universals, and sense data).
We have a form of the existence claim here, completely unsupported, but in general this description is properly hedged so that it applies to reality. It doesn't claim more than that burden of proof is found "in many situations," it recognizes that there is often a considerable amount of dispute as to who actually has burden of proof, and it notes that burden of proof is in some cases "set by the situation," which is the most notable fact about burden of proof and yet somehow not sufficiently remarked. But it's clear that these are merely the phenomena of burden of proof; they do not constitute an account, and, indeed, the constant need to hedge makes clear that accounts of the sort identified above, which can't account for all the anomalous situations that require the hedges, simply will not do.
Obviously there are cases where a burden of proof does exist. And there is a fairly obvious account that handles all the phenomena without falling into either absurdity or handwaving, and that is that the obligation of burden of proof is not something that accrues automatically. It is created, constructed, made, either by some competent authority or by the agreement of parties, or, in other words: by rules, laws, or decisions legitimately imposed by legitimate authorities, or by contract. I've talked about this before. This position has all the strength of presumptive reasoning accounts, but evades their problems by recognizing that the presumptions in question must be shared, and looks at how shared presumptions come about (i.e., by authority or by agreement). We see from this why it is so messy.
We also get from this why people spend such an extraordinary amount of time talking about burden of proof rather than giving arguments: they are negotiating their obligations to other people, not, as they often pretend (as part of their negotiation), merely identifying obvious facts. The reason people argue so much about burden of proof is the same reason that people argue so much about what they will put into a contract or treaty or agreement. But there is no way around the fact that, when there is no authoritative imposition, this negotiation is necessary for any obligation to arise -- and that the negotiation is two-way.