But isn't the goal of the t-shirt a gift for the first 20k paying users? The goal for them doesn't seem to be about what they get back from giving away the t-shirt.
Based on their public data, they have 20,000+ paying users, and 1500 users paying for the family plan, which if all of the users are on the cheapest $5/month plan (which, many probably are on the $10/month plan like myself, as that gives you unlimited searches) would give them $100K+ each month.
Sounds pretty sustainable already to me, unless they have a really large team and/or really inefficient infrastructure, neither which seem true to me.
The expensive parts of the infrastructure are not their own, and the expense will scale by number of requests.
Back of the envelope:
The Bing API costs about $0.015 per query at the most cost effective tier. Kagi's highest revenue per query pricing is $5/300 queries = $0.166 / query. That's pretty slim margins, and only made worse by the $10 tier.
Now, they can of course start doing exclusively their own crawling, indexing and ranking and avoid paying some larger search engine for API access. But for that to be cheaper, they need orders of magnitude more scale
Kagi has a bunch of different sources, it doesn't seem like all results come from the same source, so without any particular insight into how each query works behind the scenes, I'd gather it be very hard for any of us to come up with an average cost per query like you tried to do.
Sure, that's why I explicitly stated it was a back of the envelope calculation. It wont be right to the penny, but it will certainly have the right order of magnitude.
I'll assert that even a back of the envelope calculation is more useful than your initial "they must be sustainable unless the infra is inefficient, and it feels like it isn't" statement. Do you have an actual number rather than feelings? Is that number significantly lower than this estimate?