Book one night on Airbnb, then make a deal with the host to book directly. Any downsides?

I would do this only for subsequent stays and with a written contract.

I've done this. I was renting a furnished apartment for about 5 weeks over Airbnb¹. After about 3 weeks, the host asked if I wanted to renew, and if so, with or without Airbnb. He wrote a contract which we both signed, only then I paid, and less than through Airbnb.

I don't care whether or not this is against the T&C of Airbnb or not, for Airbnb as a business certainly doesn't care much about their societal impact. If Airbnb wants to enforce that if I ever rented from someone through Airbnb I may never again rent from them through any other means, good luck to them (imagine if a dating site would try to prohibit couples who met on their platform from communicating outside of it ;-). The only practical problem for me was that Airbnb wanted me to write a review when I was actually still staying there, which limited how honest I felt I could be about the property, and in fact the host was unhappy that I also mentioned negative points.

¹^{Normally, I avoid Airbnb like the plague, for I strongly detest their city killing and lawlessness akin to Uber or Deliveroo, and indeed I have never used them and will never use them where alternatives exist. In the four cases I've used it, it was very much an action of last resort as nothing else meeting our requirements was available.}

Airbnb would almost certainly consider this to be a terms of service violation. From Section 14:

In connection with your use of the Airbnb Platform, you will not and will not assist or enable others to: [...] use the Airbnb Platform to request, make or accept a booking independent of the Airbnb Platform, to circumvent any Service Fees or for any other reason;

Airbnb might come after you and/or the host, via lawsuit or arbitration or other legal means, for the fees they didn't receive. They might also ban you from the site.

Yes, you are likely to find hosts that will agree.

You are also likely to find a host who, upon your revelation of dishonesty, refuses to rent to you and reports you to AirBNB.

Keep in mind that if you e-mail the agent for the property (owner or otherwise) saying, "I'll give you $1000 to stay here from June 1st to 7th this year" and they reply, "Ok," you now have a legally binding written contract. The risk here is not about not having a contract (even an oral agreement can be legally binding if correctly formed) but in how you deal with enforcement if one party feels the other party didn't fulfill the terms (both explicit and implicit) of the contract.

In theory if you go through Airbnb they will provide both better-defined contract terms and support for enforcement. In practice the level of support for enforcement you'll get from them varies because Airbnb has conflicting interests here: one is to keep the system running smoothly but the other is to spend as little time and money as possible dealing with problems. Airbnb has in the past ignored scams until they experienced enough reputation damage from articles by journalists that they felt they had to address them.

It's not really possible to provide specific advice on how risky your idea is for any particular rental. The prospect of enforcement through Airbnb depends on the nature of the problem and how publicised it's been at that particular time, and the prospect of enforcement through other actions (courts, other governmental authorities, NGOs such as the BBB) also varies widely depending on the country and province in which you reside and in which the property is situated.

Am I likely to find hosts who would agree to this deal?

Sure. I've done it myself.

But on the part of the host-- long-term, off-the-books rentals are not wise.

The laws vary by state but if a host allows a guest to stay in their home long enough (California is something trivial like 2 weeks), and they are collecting the equivalent of "rent" from the guest, the guest gains tenant's rights in the host's home.

AirBnB themselves warn of this.

It has happened before.

If the guest becomes a nuisance or stops paying, the host cannot legally get rid of the guest without going through the eviction process, which is both costly and time-consuming. If the guest stops paying during this time, the host is denied a source of income for the duration of the proceedings while the guest extracontractually occupies the room. It is a nightmare.

Conducting your business through AirBnB at least provides some contractual boundaries to the agreement that might supercede tenant law in favor of the host.