my likelihood is dominating my prior [not!]

An interesting misconception read on X validated today, with a confusion between the absolute value of the likelihood function and its variability. Which I have trouble explaining except possibly by the extrapolation from the discrete case and a confusion between the probability density of the data [scaled as a probability] and the likelihood function [scale-less]. I also had trouble convincing the originator of the question of the irrelevance of the scale of the likelihood per se, even when demonstrating that |𝚺| could vanish from the posterior with no consequence whatsoever. It is only when I thought of the case when the likelihood is constant in 𝜃 that I managed to make my case.