Brindenstine has talked about having corporations sponsor missions before (e.g. the Doritos® ISS Crew Resupply Mission or the Mars Candy Company Rover). However there hasn't been much movement on that front, and I doubt that NASA's sponsorships would be as profitable. Imagine if IBM sponsors a rocket that self destructs after a computer glitch— it would be awkward for them and not good marketing for Watson.
The Olympics are also planning on making a billion dollars or so from ticket sales. I think NASA charges $50 or so for tickets to watch a launch at the cape. At the Olympics, some tickets may be that cheap or cheaper (e.g. the Slovenia vs South Sudan handball match, no offense to either of those countries). However tickets for more popular events like the opening ceremonies are going to exceed $1000.
The Olympics also has more events within a single year than NASA. There may be a few dozen launches a year, and they're hard to sell tickets for since most launch attempts get scrubbed/rescheduled at least once and they're not always in accessible locations. Meanwhile, the Olympics will have hundreds of events and you can be reasonably confident that they will happen on the same day that's on your ticket.
We're not leaving this planet for a long time yet. Even if we do, it's not desirable to abandon Earth, it's vastly superior to all other practical options we have. We're going to have to continue to try to co-exist and avoid any further world wars. $4b or $5b per year for a large gathering of the sort the Olympics represents, is dirt cheap if it lends even a tiny boost to global peace between nations.
Keeping in mind this isn't actually an either or situation. It's obviously not either NASA's budget, or putting on the Olympics at a $20b cost. We - humanity - can in fact afford to do both.
I'd vote for reducing NASA's annual budget by 19% (~$4b per year) to pay for the Olympics, if that was the only option. That's a very easy decision.
It's one of the few things that most of humanity does together. Over a billion people watch some part of the Olympics globally. It's a modest, important pillar of peaceful co-existence between nations and peoples broadly. We shoud do more things together in that spirit. Maintaining peace between the 195 nations is more important than NASA's continued operation, by several orders of magnitude.
For another example, the UN is also more important than NASA (the UN has a core budget comparable to an annualized cost of the Olympics). I'm not much of a fan of the UN, and I still recognize that it serves an important purpose of promoting cooperation and discussion between nations and would be almost entirely impossible to replace today.
Competitive sport is critically important to humanity and arguably always has been in some form or another. Sometimes people that don't love sport generally, entirely fail to understand what it accomplishes and how important it is.
If NASA goes away (and yes, it would obviously be a huge loss), a dozen other countries will continue with various space efforts. How many things do that many nations cooperate together on in an overwhelmingly positive spirit other than the Olympics?
Rockets don't make people punch other people in the face. Sports really do.
Rank and file people seeing the world come together and work towards a common goal, even on something as "trivial" as a game, humanizes those different people. And yes, it sometimes results in an occasional fight, but it's much better than the kinds of fights that occur which involve rockets.
Sexism? I hope you do realize that you are actually defeating your own argument here, since women are segregated and compete separately.
Trans-men, who were not even top athletes among biological men, already dominate biological women's competition.
* two people of different ethnicities holding hands
* cold fusion
Need I remind you we discovered fusion to kill people?
$20B would put an American on the moon and maybe even put an American on mars, surely that would get the world's attention.
Want peace? Let's detonate a nuclear bomb in space and get people to recognize the destructive potential of our current arms.
The olympics is designed to get people to tune in, NASA is set up to create knowledge. Judging both organizations by outcomes that only the olmpyics is concerned about is a great way to obfuscate NASA's contributions.
IMO, that's an argument against Olympics, not for it.
I'm far from calling myself an environmentalist, but consider all the waste and pollution produced by people flying in from all over the world. The negative impact is significant.