I don't think my example price of $1400 to $1500 would be too high for an M1 Pro Mac Mini.
Agreed - the existing Intel mini is already $1299 with 16GB RAM (so far the minimum seen on an M1 Pro) and you wouldn't have to argue too hard to claim that it was more comparable to the $1499 i7 option.
OTOH, so far Apple haven't use Apple Silicon as an excuse for huge price hikes, at least on the base model at each level: the only really significant change was from the "high end" 13" MBP to the 14" MBP, but that can be justified in that the M1 Pro 14" is now much closer in specs & performance to the 16" than the 13" was.
Plus, in the past computer prices have been pretty much immune from inflation, staying roughly the same price in figures, while specs rose dramatically, I think that we're probably going to have to start sucking up price rises across the board, and not just Apple.
That's basically what my last sentence said (with respect to the Intel models though). Apple would have to price a high performance Intel based Mac mini so high that it wouldn't be competitive at all as a desktop. An Intel NUC extreme barebones kit costs $2k for example, and you still have to add a GPU, ram and SSD.
Realistically, though, the Mac Minis have never been competitive with PC desktops, and Apple pulled a really fast one in 2018 when they basically switched them from mobile processors to significantly cheaper (for the same/better performance) desktop-class chips and still raised the prices. The "innovation" was really that throwing out the spinning rust and optical drive without shrinking the case freed up enough space for a desktop-class heatsink and fan. $799 for an desktop i3 with (feeble) Intel integrated graphics and 8GB of bog-standard DDR4 RAM was an absolute joke by PC standards. If using Windows or Linux was an option for your application, the only reason you'd take a second look is that it was quite sleek and lacked the huge power brick that you get with most NUC-type PCs. Of course, if you wanted/needed MacOS it was the only game in town.
As for the "NUC Extreme" - I think that is a rather cherry-picked example. It's really not in the same "mini" class, meaning it can take a full-length desktop GPU (...which is the M1/M1 Pro's weak point - they can thrash any sort of integrated/on-package graphics that would fit in a slim laptop, but their ace is performance-per-watt, not raw performance) and it's real niche is for gaming LAN parties (hence the lightshow) which is virtually nonexistant in the Mac world. Probably makes a noise like a 747 landing, too.
There are lots of smaller/quieter "mini PCs" for more modest prices, not to mention bog-standard mini-towers that can be hidden under the desk where their size doesn't matter.
Also - you've got to look at "total cost of ownership" - the NUC Extreme uses standard RAM, 4 slots for standard M.2 storage, you can change/upgrade the GPU and there's even the possibility of future "compute element" updates. As soon as you start adding sensibly-priced 3rd party RAM and storage rather than having to rely on Apple's pricey BTO options (...and the ability to add 3rd party RAM to the mini will be going away with M1) then the starting price difference will rapidly fade away.
...before this descends into PC vs Mac wars, my point is that Apple has never shown much concern about competing with PCs on price and that many Mac customers will value the MacOS compatibility and aesthetics enough to pay a substantial premium for Mac hardware. Conversely, in recent years there has always been the option to pay Apple prices for PC hardware by going with MS Surface, Razer or suchlike (and maybe NUC belongs on that list) but that isn't really representative of PC prices.
NB: you maybe overcooked the price of the NUC extreme a bit:

Intel NUC 11 Extreme Beast Canyon Series Mini PC
Intel NUC 11 Extreme Beast Canyon Series Mini PC
I think the NUC Extreme (without the go-faster lights and skull) is closer to the sort of thing we might fantasise about from the "Mini Mac Pro" - but I suspect that will cost a lot more than $2000.