I'm thinking mainly of, for example, Nuwa and Fuxi being Bodhisattvas, Laozi and Confucius being Bodhisattvas, the great antiquity of Buddhism in China(time of the sage kings, relics of Ashoka etc), Laozi being a disciple of the Buddha, Laozi being Buddha, Yin Xi being Buddha, Laozi imposing monastic celibacy with a view to genocide etc. Were these ideas and beliefs largely restricted to religious polemic, rhetoric deployed only in specific circumstances and for specific purposes, or do we see them held and used more widely? Are they known or believed by those without a dog in the fight - those not strongly aligned with either Buddhism or Daoism? Thanks
The debate you are talking about largely happened within the context of Daoist and and Buddhist Priest completing for imperial patronage during the Tang Dynasty.
There is a trope in Western European mythological fantasy of the existence of a "court wizard" who was a key adviser for the king and used his magical powers to assist the kingdom. Examples might be Merlin or Gandalf. Believe it or not this arrangement actually existed in ancient China. Imperial office holders had a staff of spell-wielding priests that lived in the imperial building and engaged in magical efforts on the leader's and realm's behalf. The early success of Buddhism in China was in no small part because individual Buddhist priests impressed an emperor with their magic and became officially sanctioned.
To bring the lens out a bit further, a big ongoing controversy during the Tang Dynasty was the place of Buddhism in Chinese Society. Buddhism became extremely popular at all levels of Chinese Society and Confucius and Daoists engaged in a multi-century battle against Buddhism. The basic arguments against Buddhism were:
From these battle lines there was a back and forth battle between Daoist and Buddhist priests for Imperial patronage. These debates would occasionally lead to society-wide persecution of Buddhism. Where this intersects with your question is that part of the Daoist strategy within these debates was to co-opt some parts of Buddhist mythology with the aim to show that Daoism was actually a superset of Buddhism and preceded it.
A good source for more of this type of information is this book: https://www.amazon.com/Buddhism-Cambridge-Studies-Literature-Institutions/dp/0521103487