Headcanon: He definitely was brave enough and engaged in it too, by necessity of Jedi being called on for mediation and diplomacy as well as the need of a general to coordinate with the civil leadership of planets he’s fighting on or for.
Downplaying his interest in politics would be both a sign of humility (“I don’t need to rub elbows with important people, I’,m just a humble guy”) and sensible politics in itself by signalling that he’s not out for power plays, just doing his duty.
That also kinda tracks with Marcus Aurelius, who allegedly didn’t like being named Emperor by the senate as he preferred philosophy over power. He did accept the office out of a sense of duty, on the condition that his adoptive brother be made Co-Emperor.
The parallel I’m getting at here is that both might not have been passionate politicians, but will have competently and dutifully risen to the task when called on. They were also both militarily successful in the later years of their respective careers as Emperor and as Jedi Master.
Hence: Marcus Aurelius Antoninus an Obi-Wan Kenobi are the same person. QED
I get why it’s done from a narrative point of view, but some obnoxious pedantic voice in my head never stops wondering why people would write down obscure lore notes in the middle of a disaster or when they’re about to die. It makes sense for some organisation’s agents to document their discoveries and leave (incremental) reports on-site in case they don’t make it back for a full report, but some random raider in FO4 noting down how pissed he is with his boss?
Of course, I’m a sucker for lore, so I’ll take it, particularly when it’s in audio log form during exploration sections so I can keep exploring and listen instead of stopping to read.