This is an inherently subjective question about how things should be rather than how they are, so I don't think a definitive answer is possible.

It is at least a very interesting question, since it gets at two problems for the price of one. The first is obviously one of gatekeeping - how much gatekeeping happens in history, and how much is necessary. The second, less obvious problem is 'what is a professional historian anyway' - because if we can't decide that, then even if we want to gatekeep then we don't really know where to draw the line.

I got into this second issue here here a little while ago - namely, laying out my view that drawing the boundary of who a historian is by either qualification or professional status (ie someone who gets paid to do history) is messy. Excluding those without (say) history degrees from being considered a historian is highly elitist, and would exclude any number of people who contribute to our understanding of history, and include any number of people who don't. Considering those who are paid to do history isn't much better - particularly in today's academic climate it's at best a messy measure given how many great historians are precariously employed, and will always exclude a lot of voluntary work, and include work (like Irving's!) which is problematic but makes money.

The measure I settled on in that thread is that the best metric is "are you creating new historical knowledge". That helps us draw the line between historians and what we might term history popularisers or communicators, who share their (often considerable) knowledge, but aren't engaged in the methodological process of discovering new stuff about the past.

Broadly speaking, scholarly publishers are already set up to gatekeep along these lines. To get something published in an academic history journal it doesn't just (or even) need to be well-written or convincing, it needs to make a defined, structured contribution to existing knowledge. Peer review exists to ensure this is happening - contrary to popular understanding it doesn't exist to fact check everything in a piece of writing, but rather ensures that it is methodologically sound (ie its conclusions are supported by appropriate evidence) and that it's making a significant contribution to the field. In theory, someone who doesn't work at an academic institution can submit their work to such an outlet and have it published if it meets these standards, though in practice relatively few do because it's a thankless task with limited payoff unless your prospects for employment/promotion rely on it.

In contrast, popular history faces far fewer such boundaries - limited or even no review for quality, rather an assessment of whether it will sell or not. You can just repackage existing ideas or retell known stories ad nauseum so long as people keep buying them (certain genres of military history lend themselves to this). Accuracy or credibility aren't completely irrelevant though, since publishers do care about their reputations. If the reputational damage of publishing a bad book outweighs the commercial benefit, then it won't get published (though it's fair to say that not all publishers care equally about their reputation, nor are they equally good at judging the risk of damaging it).

These two mechanisms are, broadly, fine as they are in my view. Popular historians serve purposes that scholarly historians don't - writing for a broad audience is a different skill to doing the kind of painstaking analysis historical research is often characterised by. Some historians can do both at once, but many can't. There's room for people whose job it is to synthesise and popularise knowledge, even if what they write might sometimes be regarded as "bad" history by scholars. Rather than stopping it altogether, holding publishers to account and making them think about their reputational risks is a much more proportional approach.

This brings us back to Irving though, because ultimately what Irving was doing was not writing "bad" history. Irving was (or is) doing the opposite of history writing in his work. Historians will always have different interpretations and come to different conclusions about the past, because this is always subjective. But we are always working from that evidence - the fundamental starting point of the historical method is that we learn about the past by building meaning on the evidence we've been left. Denialism works the opposite way - it is so named not because it denies a historical interpretation, it denies the historical record itself. Irving's conclusions are not possible without denying the validity of vast swathes of evidence about the Holocaust, or relying on twisting what he does use. It is not an effort made in good faith to better understand what happened in the past, but rather a deliberate distortion of it.

Publishers have no business publishing denialism, about the Holocaust or anything else. If they do so, it is grounds for authors and public alike to boycott them. Some countries have banned some specific instances of it, most notably Holocaust denialism across much of Europe. It's not a question of qualifications though - it's a question of purpose and method. Making David Irving do a history degree would not fix his work, it would merely lend it a veneer or credence it does not deserve.