More generally, what is the nature of the gamut? Primaries as peak sensitivity don't behave the same as primaries for mixing output.
I suppose this is necessary to know for RAW importing and would be explicitly stated in a DNG file. Do they vary greatly between cameras?
The output from the sensor is RGB based, so if interpreted simply as three primaries its gamut would be triangle shaped as you are familiar with for other RGB gamuts.
Of course as you allude to each of the three colour filters in the sensor will not only represent a single wavelength but a curve, so for accuracy a colour profile needs not just primaries but a colour look-up table.
The colour responses vary greatly between cameras, enough that you need an input profile specifically for the camera. Your RAW conversion software will have input profiles from major cameras built-in and may even have a mechanism for automatically downloading new ones. In some cases it may not have a full profile but instead just have a matrix (eg just a linear conversion from primaries to CIE XYZ) and work from that, with less accuracy. And if you don't have a camera specific profile or matrix, the colours are likely to look wrong.
I'm wondering if the reason Lightroom now uses ProPhoto is because the primaries (and thus the native gamut) are outside of AdobeRGB?
Yes, a lot of cameras extend outside of AdobeRGB, but that's not the only reason for ProPhotoRGB.
ProPhotoRGB is seen as a good default/universal "working" colour space. The thing about a working colour space is that you apply filters to image data after it's pulled from the camera, that may even further boost colour saturation or shift colours around, so from that point on the gamut from the camera sensor's output is irrelevant, and you want something that is big and generic enough to represent any colours you're creating so you don't have to worry about colour clipping even if you do something crazy like boost saturation a lot then reduce it again. In short ProPhotoRGB gives you breathing room as a "working" colour space but your image will of course be reduced back down to an output colour profile for viewing on screen or exporting (unless of course you're keeping the wide gamut when exporting).
Conversion between these colour spaces requires the sort of mathematics you'd do in the first year of a math degree, suffice to say that most conversions from colour space A to colour space B go from A to CIE XYZ first, then from CIE XYZ to B. CIE XYZ isn't practical to use as a working colour space itself due to its inefficiency and that applying filters is simpler in an RGB domain or a domain based on luminance.
I don't know how well I've answered your questions.