We spent a week in Bhutan earlier this year, and could not have enjoyed it more.
After much online research, we identified an independent tour company which offered a driver, guide and people carrier for 8 days for U$100 per day. We then organised all of our own hotels through WhatsApp. Regrettably we weren't travelling on an Amankora budget, but every hotel we stayed at was excellent (hospitable staff, clean rooms and delicious food). Having our own driver and guide was ideal - we set the schedule every day, made decisions as to what we should do, and they complied without any issue.
It's also possible to explore Thimphu and Paro independently (we took advantage of doing so on several mornings and evenings).
Given the mountain roads, and the relative compactness of the country, it's unclear what the benefits of non-guided travel would be (and this is said as someone who had only travelled independently before this Bhutan trip).
Additionally, Paro airport is a unique experience. Landing from the south and looking out the port side, the immediate reaction is 'crikey, we're close to the side of the valley here'; you then look out the starboard side and the side of the valley is about one-third of the distance away!