I disagree completely, because there isn't a mixed message when you realise that the inconsistencies are about characters who joined the team much later in the comics.
There is only one reference to the start or beginning of the JSA and that is the one where the Dr Midnite AI shows Charles Mcnider's bio. The only other reference to the teams time line is their demise in 2010 which was confirmed by the bio.
All other historic references are to JSAers who weren't original or early members of the team. In the comics Wildcat was only first referred to as a working alongside the JSA (Not a member) at the end of the GA. He made a guest appearance in issue 24 in 1945 (same issue the original Mr Terrific joined the team) and also guest appeared in the last 2 issues of 45. He was finally an official member in the JLA Cross overs of the 60s.
SSK (Sylvester) didn't join the JSA in the comics until All Star comics were revived in 1976.
References to Sylvester in his youth and Ted Grant as a boxer don't have to reflect the time the JSA started because they weren't a part of the JSA at that time. In the comics Dr Midnite didn't join until 1941 but the team started in 40. Midnite joined after GL left from getting his own self titled non-anthology (same reason Jay Garrick left when All- Flash came out and was replaced by Johnny Thunder) and hourman left due to declining popularity (which was why Ted Knight Starman joined at the same time).
So them being vague has more to do with the fact that their time of joining innthe comics was vague, it happened between publications. They weren't a member here but were next time they were published. If anything it's actually consistent for that reason.
On the wiki for the show
https://stargirl.fandom.com/wiki/Jus...ety_of_America
It says the JSA started in 41 with Midnite. It refers to Pat and Syl as 90's heroes. It says that Wildcat and Hourman joined the team at a later date (presumably to have it consistent with Rex being young enough to have a teenage son in 2020) etc completely in line with these 'inconsistemcies" and mixed messages.