After doing a bit of research, it seems like pure/raw tung oil might do the trick.
I'm going to discount its suitability for this below but I want to caution to be careful of user reports of tung oil's superiority to BLO as a finish because, to be frank, a lot of them are accepting its reputation on face value and have never given it any sort of proper testing1. Its reputation for superior waterproofing can be wildly exaggerated, and like any straight oil using in finishing there's a limit to how much heat it can resist (all cured oils soften noticeably when heated, e.g. from the bottom of a hot coffee mug).
at least somewhat water resistant (don't need to worry about using coasters)
If you genuinely want to go coaster-free this is quite a bit beyond somewhat water resistant and actually discounts all penetrating oil finishes2.
You almost have to use varnish or lacquer. Only those are sufficiently heat- and water-resistant that you can use the surface with complete impunity from water marks, coffee stains and heat damage.
And in terms of consumer-level finishes you're really looking at polyurethane.
For the 'natural' look and feel as specified it's best to use it in the form of wiping varnish, which I was going to recommend anyway because it's far easier to achieve a good result with varnish diluted to a thinner consistency than it is using it at full strength.
See more on wiping varnish in this previous Answer.
can I go with something like polymerized tung
Polymerized tung oil is greatly improved as a finish over basic raw tung oil. However it can be extremely expensive, and on top of this requires a very slow and labour-intensive finishing regimen that makes it much less suited to large flat surfaces like a tabletop.
As a side question, is BLO food safe? I have heard Ikea claims their Behandla finish is food safe, but the listed 'metal drying agents' make me skeptical...
You're right to be sceptical but the short answer is yes.
I need to mention that there's a lot of disagreement about the question of food safety in relation to finishes in woodworking circles online but the basic fact is that all finishes should be considered food safe, especially on something like a desk. However even for food-prep surfaces every conventional finish, including those that contain heavy-metal driers, is safe once fully cured as far as we know.
Here is Bob Flexner on the subject from one of his books (my emphasis):
Food safety is a non-issue because there's no evidence of a problem and there never has been. So far as we know, all finishes are safe to eat off of, and safe for children to chew on, once the finish has fully cured (the rule of thumb being 30 days).
I want to make clear that I'm not saying that all finishes are food safe—we can't be absolutely sure about the safety of any curing finish. I'm saying that there is no evidence of any common wood finish being unsafe for food or mouth contact once it has fully cured, so a distinction between food-safe and non-food-safe is speculative.
That's as unequivocal as I need a source to be, but others reading the same thing retain their doubts. So, as always, you have to make your own mind up about this sort of thing.
1 The fact is that many users aren't really subjecting surfaces finished using tung oil to much hard use, sometimes because of the effort it has taken to get a good result with it in the first place! Tung oil is noticeably slower to finish with than BLO, only partly because of its much slower drying time.
2 You can achieve a respectable level of protection from various oil finishes but only if you're willing to spend a month or more doing the finishing job..... yeah, didn't think so ^_^ By comparison, it'll often take less than a week to achieve a superior finish using varnish, plus necessary curing time after application.