(This is well outside my area of expertise. To the extent I have a comparative advantage in covid-y matters, it is almost entirely in medium-term predictions <6 months out. I am also entirely thinking of "biosecurity" as preventing pandemics as bad, or up to 1-2 orders of magnitude worse, than covid-19. I don't have a strong sense of what preventing existential GCBRs will look like, though I would love to learn, and I imagine there are probably many subtle considerations that I have missed)
The framing of this question suggests to me that you are looking for policy failures from overreacting to covid-19. I think this is a very serious concern.
However, one thing I want to flag is that it's not obvious to me that failures from overreaction are necessarily worse than failures from underreaction. And I'm personally roughly 50-50 for this proposition. (EDIT: 2020/08/04 I think I'm currently closer to 60-40 that mistakes from underreaction is likely to be worse than overreaction)
Several considerations for me in favor of thinking of underreaction being the default:
1. Early on in the pandemic (say before mid-late March), I'm aware of many groups underreacting (basically all of the geographical West, including Latin America). There are some governments and institutions that reacted well (shout-outs: Mongolia, Seattle Flu Study), but I'm not aware of any institutions severely overreacting early on in the pandemic, other than a) maybe tiny groups like the Bay Area rationalist community, and b) debatably a few national governments like India (even there it's not clear to me that it's obviously worse than underreacting).
2. At the high level, I think what happens is something like regularization: governments and other institutions by default have something like a "target reaction level": This cause them to overreact to small events but underreact to large events. Since small events happen much more than large events, this will cause them to appear to usually overreact, so the received wisdom becomes that governments overreact. However, when reactions really matter, this will usually mean that they'll be underreacting.
(Note that this is subtly different from claiming that this is an unavoidable error in that in the absence of clairvoyance, you always either underreact or overreact; Instead, I claim that even in situations where something is predictably much worse than anything you've seen before, you, institutionally, will still be biased towards underreaction).
3. The history of pandemics (at least for events large enough to enter a casual reading of the "history of pandemics") is mostly one where governments institutionally underreact, eg, by focusing on assuaging fears rather than a) honest communication or b) focused entirely on solving the object-level crisis.
4. While it is not exclusively the case, a lot of the academic and political Twitter that I follow appears to either a) be counterfactually fatalistic and think we couldn't have done much better or) be playing a "blame game": where the bad reactions are blamed on specific politicians or officials. If a) is the case, then we did the best we could and we should be worried about overupdating on this pandemic and thus making worse decisions in the next pandemic. If b) is the case, we likewise should not change much except for which leaders we elect or otherwise select to be in charge of public health issues.
However, I find this dubious because I think most of the West (geographically, not culturally. New Zealand seems to have done fine) has done substantially worse than East Asia. While the decisions of specific politicians and bureaucrats certainly exacerbated this pandemic in predictably poor ways, the fact that most of the West has not handled the pandemic well means I don't think it's fair to pin most of this on individuals. Instead, it might be instructive to (possibly in private) consider institutional and cultural issues.
With all that said, why do I think underreaction is not more likely than overreaction?
1. Most of all, this just seems like a really hard question that I feel woefully ignorant on. There are a lot of important considerations I haven't thought through in detail (and many I probably didn't even consider)
2. It intuitively makes sense to me that governments (especially elected officials and political appointees) will overreact/overindex on the particulars of a specific past crisis. A common saying is that "generals are always preparing for the last war."
3. Even if you buy my argument that gov'ts and institutions underreacted in the beginning of a pandemic, it does not necessarily follow that they will underreact at the end. An analogy I like is to consider an inexperienced driver. The driver might be slow to realize that they should stop, but by the time they do, they will rapidly slam on the brakes even when it's unnecessary to do so.
I studied the backside of pandemics much less than I studied the beginnings of pandemics (and even then it's quite limited). For future work, it might be helpful to a) look at historical examples of recoveries after pandemics and other major crises, and thus form a well-informed outside view of whether we'll more likely to expect underreaction or overreaction and b) carefully consider the details of specific proposals in question to think through whether they help or hinder biosecurity and other health efforts.
This answer should be seen as a general case for agnosticism and fighting against potential biases towards conservatism, rather than positively advancing a case for any specific predictable policy failures from underreaction.