This seems analogous to the other sort of budget.

Suppose I decide that I'm allowed to spend £500/year on books and no more. This isn't because once I've spent that much I get no further value from them, and it isn't because if I spend more then I will literally run out of money and starve. It's because there's some degree of diminishing returns from buying more books, and there are other things I want to be able to spend money on, and putting a hard limit on how much I spend is a nice simple system that approximates the Right Thing. It doesn't approximate it very closely, but that's OK: the difference between approximating it very closely and approximating it rather crudely is small.

(I don't, in fact, put a ceiling on how much I'm allowed to spend on books.)

So, how does this relate to Covid-19 risk budgeting? I think the key claim I want to make is that I don't believe you when you say "you could instead imagine a variety of different activities [...] that do add up more or less linearly". I mean, of course you could imagine that there is such a variety of different activities, but I don't believe that you can imagine a specific set of such activities that actually do add up nearly linearly. At least, not when things are fairly bad Plague-wise.

The reason is that (so at least it seems to me) social activities (which are the things people bother with microcovid-budgets for) typically bring positive utility in two different ways. First, the fun of doing whatever thing it is: the pleasure of eating well-prepared food, the enjoyment of walking somewhere with nice scenery, the appreciation of the music at a concert, etc. Second, the extra gain of doing it with someone else. And, while some people get more out of being with other people than others do, I'm pretty sure that gain doesn't scale up linearly with the time you spend with those other people. Not even if you're doing a variety of different things with them. And when the Plague is bad, I think that (at least for the sort of person who is likely to be drawing up microcovid-budgets in the first place) there isn't enough of the first kind of utility to justify the risk: that is, the expectation in the OP here is consistently negative. But there is, at least sometimes, enough of the second, because for most people going completely without human contact is terrible. But that one has diminishing returns.

Those diminishing returns surely don't look at all the way they would need to for a risk budget to be exactly The Right Thing. (That would mean something like marginal utility = constant when t<T, and zero when t>T.) But they might have the property I claimed financial budgets sometimes have: that the utility difference between having a budget and doing all the utility calculations right is small. Do they?

I think they plausibly do if three things hold. (1) That for the activities being considered, the utility gain that matters is the one that comes from doing them with someone else -- that the utility from the thing-in-itself is much less. (2) That, on the margin, this utility gain is roughly proportional to the risk, at least conditional on the way you will actually behave. (3) That the actual risk budget figure you choose is somewhere near to optimal, given that you're risk-budgeting.

If those three things are true, then 1 and 2 mean that total utility is roughly some function of total risk; if that were exactly true then optimal behaviour would be continuing to socialize until reaching some risk level and then stopping, which is exactly what risk-budgeting does; if it's roughly true then I think that means that that behaviour (with the correct threshold) isn't too far from optimal; and then 3 means that you're near to the correct threshold.

I think 1 is plausible. (Because for many people the gain from having some social activity is very large, and microcovid-budgeters are generally being cautious enough that they have rather little social activity.)

I think 2 is plausible. (Because if you're microcovid-budgeting you will be picking your activities and how you do them so as to maximize social-utility-per-unit-risk, which means that most of the things you do will have about the same social-utility-per-unit-risk. Also, for obvious reasons there is a correlation between the two -- e.g., the more people you're with, the higher the risk but also the more social interaction you get.)

I think 3 is plausible, but this is the place where I think things are most likely to go wrong. (Not least because I don't have any argument for its plausibility that I really trust. But I think I think that while people are terrible at judging risk, it's mostly the actual risk estimation that's bad rather than the ability to make reasonable judgements of what utility gains are worth what risks.)