Aside from not being a probability in the common sense (i. e. not concerning the odds of events), as far as I understood it, the "market's attitudes towards risk" are actually factored into / built in the "risk-neutral probability". For pricing we do not have to further discount the expectation value taken according to the risk-neutral measure.
So why is it called risk-neutral then? Why not "risk-observant" or the like?
Originally "risk-neutral" is a term from economics describing the attitude of investors towards risk: if they are risk-neutral they only factor in the expected value of a decision and not the level of risk. "Risk-averse" would mean that they prefer investments with lower associated risk ceteris paribus (i.e. all else being equal).
In quant finance risk-neutral basically means the same: because risk is assumed to be hedged away you don't have to factor it in to e.g. price derivatives.
You can think of it like this: if you were to live in a world of risk-neutral investors you could use risk-neutral probabilities right away to arrive at the same formulas to price derivatives, without any transformation from real-world probabilities.
