Romans were nosy types, and rich Romans were always interested in whatever weird new stuff was going on. Some of them were extremely nosy, like the Pliny family.
Romans took court cases pretty seriously, because Roman justice was one of the things that built their society and their empire. Every Roman man who wanted a government career had to serve in various offices, and a lot of them were involved with court cases. So a guy like Pilate, who was at the top, would already have done a lot of legal jobs at the bottom.
Court cases were also something that Roman men liked to attend, as a way of passing time. An interesting defendant, a controversial situation, a good advocate who would give entertaining speeches and make interesting points – that was not just civic duty, but also a good way to pass an afternoon at the Forum, back home in Rome.
Also, acting as an advocate or magistrate allowed a Roman man to display his wit and wisdom. You see this a lot in records of Roman court cases. If you could make a defendant expose himself as a crook, or if you could make people attending court laugh, you enhanced your status. You were not just a rich Roman of noble family; you were a Smart Guy on the Move. Impressing your subordinates and household was also worthwhile, because reputation was important to a Roman and elicited loyalty and political support.
So yeah, Pilate acted exactly like most Romans would. (It was kind of a good day for Pilate, as he was known for sometimes being one of the Nasty Romans who killed politically restive foreign people in mass quantities. But the emperor had told him to tone that down.) He was looking for a big gesture to enhance his rep, and he was interested in the weird foreign holy man who did magical things and said wise sayings. Jesus didn’t play with him as much as he was hoping, and the anti-Jesus leaders brought up embarrassing points, so he cleared Jesus out of court as soon as he could. But he did petty things to annoy the high priests in the process, because you couldn’t be a Roman without getting something back for them annoying him.
Anyway… even after Greek was overwhelmingly the language of trade and of educated men, Latin was still the language of court cases (or at least of court records) and of law books, as well as of ethnic Romans at home and North African colonists (who sometimes spoke Punic at home). A fair number of the Greek patristic writers and Fathers also spoke and wrote Latin, because they trained as lawyers. (Berytus, which today is called Beirut, was a big town for law school and for Biblical studies. Stuff tended to mix.) Court was held in the local language, but you had to know Latin to do the paperwork. Heh.
So if Jesus had spoken Latin to Pilate, it would have been appropriate to a court case, but it also would have constituted trolling the Roman guy.