Hmmm. Once I notice it, I cannot but attend to how it changes my perception of the source.
Well this is fairly crucial to this topic. I say that I am 'hearing through the room' to the source, so that whether I am in an anechoic chamber or a church, I recognise voices and where they are coming from. I am aware of the acoustics of course, but I am separating them from the source. To me, then, room correction* would be folly: changing the source but leaving the room the same.
But you say that the room affects your perception of the source. Maybe we need a distinction: is the room affecting your perception of the tonality of the source, or at some other 'meta' level e.g. church-like acoustics imbue the source with calmness and seriousness etc.?
To me, the former seems unlikely. It would suggest that you would not be able to look at a source and determine from its materials, dimensions and so on, how it was going to sound or vice versa. Instead, it would be dependent on the room's acoustics, and even walking around the room, the source would be changing its apparent dimensions and materials as you walked! I perceive no such changes as I walk around a room chatting to someone.
In our house, if I walk from a highly reverberant large room into a carpeted smaller room carrying an iPad playing a podcast, I do notice a huge change in the acoustics, but I perceive no change in the voices, nor the iPad's own 'sound'.
*Not necessarily speaker correction/compensation, however.