I do
not believe that the unnamed epidemiologists who purportedly claim it was due to "public health measures"; if this is true then the countries with the least stringent measures are also being successful, and you'd also expect a correlation between stringency of measures and levels of flu. But there is no evidence of this.
I believe viral interference is much more likely, for example the 2009 H1N1 pandemic may have been disrupted by Rhinovirus strains:
These findings show that one respiratory virus can block infection with another through stimulation of antiviral defences in the airway mucosa, supporting the idea that interference from rhinovirus disrupted the 2009 IAV pandemic in Europe. These results indicate that viral interference can potentially affect the course of an epidemic...
So is the correct way to think of "viral interference" simply that a currently/recently virus-infected person is less likely to catch a second virus due to the body being in defense mode? That certainly sounds plausible. How long does the effect last? Is it just that during and immediately after an infection the immune system is in full swing and thus repels other infections as well (your linked paper abstract seems to mention timeframes of 3-6 days) or is it a more complex heightened defense readiness that can last for months?
Focusing on the UK, you could try to compare with the flu jab, because presumably the effect of viral interference is simply that a covid infection causes you to become temporarily immune to the flu (with some degree of efficacy in between 0% and 100%). So you could look up the estimated number of Covid infections in say the first wave (not just positive tests, which are obviously much lower) and imagine that those people in effect got a flu jab - would this be a significant increase over the number of people already taking flu jabs in any given flu season? (Does the answer depend on whether you only count people who were infected by Covid "in the last 3-6 days" or "at any point in the last 6 months"?) This should give you a rough idea of how much of an effect Covid viral interference would have had on the flu.
As for comparing between countries, as a more concrete observational approach on figuring out how large the effect of viral interference is, compared with the effects of changes in regulations or behaviour (admittedly other effects have been proposed that may also be important, but this seems to be the main ponit of contention in your initial post), the difficulty is of course that you'd expect everything to be correlated by default: More Covid cases in a country leads to all three of increased viral interference, increased government-imposed restrictions, and increased voluntary behavioural changes. So our best bet in answering the question would be to look at countries where these correlations are weakest.
For example, I think early on some countries like New Zealand, Taiwan and Vietnam were known for quickly imposing strong restrictions without having a large number of Covid cases. So how did the flu fare over there? If the flu reduced much less than in other countries, then the reduction in other countries was likely due to viral interference. If the flu saw a similar reduction as in other countries, then the reduction was likely not due to viral intereference.
It would be good to also have opposite examples, i.e. countries with large numbers of Covid cases but no change in regulations or behaviour. However, while information on regulations is straightforward to obtain, I think it would be difficult to ascertain how little people's behaviour changed - even if a country with no restrictions still had scenes of mass gatherings and packed public transport, it's difficult to tell whether those scenes remained as frequent as before the pandemic, and whether the average person might have started washing their hands more often and avoiding going out as much simply based on scaremongering media coverage (such as the images of military trucks lining the streets in Italy that were likely broadcasted all over the world).
