A decision to charge a player happens pretty early on after an allegation doesn't it? I would have thought the FA have enough material in their possession to decide if they wish to proceed with charging Firmino by now or very shortly.
Also Kenny & Lallana were all in close proximity to Madley when whatever happened off camera was said. They didn't react to Firmino's language.
A decision to charge can only be made on the weight of evidence. The evidence is presumably unclear or disputed.
As we know, Madley was right there, looking almost directly at Firmino when whatever was said was said, and we could see on the field later than he told Holgate that he didn't hear anything, which Holgate appeared to find that hard to believe. So it's an investigation, because Holgate is adamant, and also the only one who was really locking eyes with Firmino at the time.
Madley is mostly concerned with ensuring it isn't a physical stoush. He was also contacted by Lallana at that moment and turned towards him. He then looked back to see if more people were coming from that way, or perhaps the linesman was in his ear and he started turning to look that way (probably not that). He had also just run in at quick pace to get between them. It is also the Liverpool Derby, in the cup, after Firmino has been shoved him into the crowd - so 50,000+ people were yelling. Also, both Kenny and Lallana were talking to Firmino at the time. So it's believable no-one heard what Holgate heard.
Whatever. The point is that it's ridiculous to think Holgate in the heat of the moment, mid-way through trying to give his best 'hard man' look at Bobby, suddenly decides to make a very serious accusation because he wants to avoid a card; a card most refs would've given him anyway so there isn't even any logic to it. That is fantasy stuff. The sort of stuff an online troll would claim, but online trolls are FOS.
And going back to Madley, as soon as whatever happened happened, he suddenly felt Holgate pulling his shoulder and complaining about Firmino. The shoulder-pull is totally not on, but the accusation is serious. With that going on, plus everything else, concentrating on any one word that may have been said in that cacophony, while with a newly incensed Holgate yells, is going to be hard. So it's an investigation.