I still don't have a clear answer regarding the need to be accompanied by a tour guide. The department of tourism gave me answer that conflicts with the information on their own website.

I contacted all of the tour companies I could find that were offering accompanied day trips. Every one of them insisted on also selling me flights, hotels, and visa services. Even though I approached them regarding a day trip they had advertised, they all wanted to sell me a multi-day tour package. All of them wanted payment in advance with no right to cancel.

There are very few tours in Bhutan bookable on the popular tour sites such as Viator, Get Your Guide, Agoda, etc. They all have very high prices, but some are refundable. (The option to cancel is important for me given the likelihood of rain during my travel dates.)

It seems that Bhutan's tour operators are not yet tuned-in to the independent travel option, or perhaps don't want to be. I'm sure they still have enough people willing to hand over very large amounts of money, but I won't be.

They are selling standard services at very premium prices and getting away with it by telling people there is no other option. In all my communications I felt like I was being fleeced, and there were some very odd demands that I send copies of my passport and flight bookings before they would even give me a price for their tours. One company had reasonable prices for day trips on their website, but when I said I only wanted the day trip and not any visa services or flights or hotels or a personal car and guide for my whole stay, those prices suddenly doubled.

I did find out that Kona Bikes in Paro have mountain bike hire for $35 per day, though after I have booked their website has gone, so who knows.
I did find a list of DoT approved tour guides with contact details
I found two other listings of privately bookable tour guides, one that has some prices and one that doesn't

And finally I asked the hotel I've booked.
They will arrange a tour guide for BTN 2500 per day (approx USD 30), and a car for BTN 4000 per day (approx USD 48).
Those prices are much more reasonable, there is no requirement to pay in advance, and (with tips of course) I suspect that much more of my money will make it into the pockets of the guide and the driver instead of the bank account of large tour operators who are only middle-men and whose owners aren't even Bhutanese companies.

This isn't about penny-pinching and not wanting to pay for a service.
It's about not being willing to pay massively inflated prices that far outweigh the value of the service being provided and especially where it's quite clear that the biggest share of that fee is not going to local workers. Agencies I contacted only wanted to sell me packages for thousands of dollars.
It's also about wanting the flexibility of independent travel that goes where and when I want, at my pace, and suiting my interests.

If Kona Bikes still exist and if the hotel delivers as promised, then I look forward to a few hours dragging a tour guide up to the Tiger's Nest before I jump on a bike and set off to explore in freedom, unless or until some officials tell me I'm not allowed.