Then, you had lawyers, advocati, who knew the legal stuff. They did not talk in court, though.
Is this quite right? In civil cases, "court" meant two things. First there were proceedings in front of the praetor, who would prepare a set of instructions for the judge, that is a set of instructions: "Let V. V. be judge. If you find that A.A. sold N.N. a horse, then enter judgment for A.A. unless you find that A.A. behaved wickedly and fraudulently, in which case enter judgment for N.N." That "formula" would then be taken to a judge who would actually decide the case. These were literally proceedings "in court", in the forum, before the magistrates. But they did not decide the case, they simply prepared it for decision. They were not concerned to know who was right or wrong, merely what allegations each side wished to make and how far those allegations should be recognised as legally relevant by incorporation in the formula. The argument between lawyers was, in essence: what facts are to be regarded as legally relevant (if proved), not what facts can be proved.
That first stage --- the proceedings before the praetor --- is pure lawyer's law, and it would make sense to assume that it would have been handled by professionally trained technical lawyers. It would have been "in court".
The second stage, the proceedings before the judge (who would be appointed in the formula specifically to hear the case) would be in private. This is where the real "persuasion" and reference to the facts and so forth would take place. It is what a layman would most clearly recognise as the "court" or "trial". The legal issues had already been framed by the Praetor's instructions, so It would not need a law-lawyer but what one Texan plaintiffs' attorney once described to me as a "fact lawyer". This is where the real Ciceronian side of advocacy would have come to the fore.
Of course, in the absence of any formally organised legal profession one cannot say "advocati did this", "patroni did that", "jurisconsulti did the other". But I would have expected civil cases at least to require oral advocacy (in the modern sense) by specifically legal experts at least at the Praetorian stage. Criminal and public-law proceedings (which is what Cicero was normally concerned in) were quite another thing and are, I suspect, less well understood. (At any rate, I don't know much about them.)