Post-Brexit travel to France to conduct holiday let maintenance/administration

As a UK citizen then when the UK and EU transition agreement ends after 31st December 2020 then you will still be entitled to enter the Schengen area without a visa for short stays under the visa waiver program.

Short stays means that you can enter and remain any many times as you wish as long as you don't exceed being in the area for more than 90 days in an 180-day period (a rolling window), have a valid passport with > 3 months remaining etc.

This does not explicitly state what purposes you may or may not enter for (so it is not only for tourism purposes for example), so generally speaking you are allowed to do many things such as tourism, business purposes, visit friends & family, attend events, get medical treatment, etc. Maintenance of your own property should therefore not be an issue. So almost anything except Study, live long-term or work (Work meaning taking up a position in a Schengen country).

As such, then the citizenship of your partner does not really alter anything as long as you are just visiting because you already have the privilege to travel to the EU without a visa. It gives you other advantages if you want to move & live or study in the EU, or stay longer, or join your partner who is living there etc. but that is not the scope of your question.

When the EU introduces the ETIAS online travel authorisation program in 2022 then as a UK citizen you will have to apply through this website in order to obtain your Visa-Waiver approval. This will cost approx. 7 Euros and will have to be done in advance of travelling, but otherwise nothing else much is expected to change (details are still lacking, so please keep an eye on this. As noted in the comments, it is unclear if this fee might be waived for you or not.).

Being a EU family member will also not alter this. You would still need to obtain this approval before travelling. Unless you are joining your EU spouse in the EU to live, work, study etc.

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.