my spouse is a Dutch citizen and, at least under French tax law, the lettings business is treated as our joint marital income. Are we therefore exercising EU free movement rights when travelling to France?
No. If you travel to France with your Dutch spouse, you will be exercising EU free movement rights because you are traveling to France with your Dutch spouse. The fact that you have joint rental income from a property in France does not matter.
I’m pretty sure there’s some nuances to it. A non-EU person isn’t entitled to simply enter an EU country when presenting at a border with their EU spouse.
Actually, they can. They also need a passport and in some cases (not including citizens of the UK) a visa that must be granted without charge on the basis of a simplified and accelerated application. But even if they lack those documents, the directive requires then to be given the chance to prove by other means that they are entitled to the right of free movement.
Firstly, AIUI, free movement rights only apply in “host” countries—ie those other than the EU citizen’s “home” country (ie they might apply to us in France, but would not apply in the Netherlands).
This is the general rule, but would not normally apply since your Dutch spouse has been living with you elsewhere in the EU (the UK), you can actually make use of free movement rights in the Netherlands, too. This would be true even though the UK has left the EU, since it was part of the EU while the two of you were living there together. (However, you mentioned in a comment that your spouse is also a citizen of the UK, which is the exception to the exception, meaning that you would indeed be subject to Dutch national law should you want to move to the Netherlands.)
Secondly, they are only engaged if the EU citizen is exercising their free movement rights, for example by working (or seeking work) in the host country.
This is a common misconception. As Relaxed has noted in the comments, the free movement directive outlines a right of entry, a right of residence for up to three months, and a right of residence for more than three months. Only in the last case can a country require you to show that you are working, studying, or in possession of sufficient resources.
Thirdly, non-EU family members require documentation that I do not have.
What documents? The only document required by an Annex II family member of an EU citizen is a passport, which I suspect you have.
Fourth, with that documentation the non-EU family member should be entitled to travel whether together with the EU citizen or separately. So presumably, if this applies to us, I should obtain some form of documentation from France that recognises my status?
Unfortunately that's not how it works. If you actually establish residence in France together, you can get a carte de séjour, a residence card, which would allow you to travel in and out of France independently, but if you do not do that then your independent travel to France will be as an Annex II visitor.