Don't forget, a referee must send a player off for "using offensive, insulting or abusive language and/or action(s)". The referee cannot have heard anything untoward, or he would have sent Henderson from the field. The allegation must have come from the Arsenal player to the referee, and the referee then reported what had been said to him to both managers.
Any player hearing the allegation made by the Arsenal player to the referee then relays the allegation to his teammates, and hence the reactions from Firminho, Allison etc.
Whether anyone else heard Henderson use racist or abusive language towards another player is unknown. However, it appears that nobody testified to this, and nothing was picked up by TV cameras.
On this basis, the FA really has no option other than to drop the charges against Henderson, because it is one person's word against another's. This isn't really a case of the FA succumbing to LFC pressure. Henderson at least knows what was actually said though.