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So a bretonnian baron has the same stats except 1 extra I for the GM. Then does the GM always win that fight (on average) because he always goes first. If this is the case then the 1 extra I is worth the price of the Baron. In this case 100 points.

Flawed logic, you can increase any relevant stat by 1 (be it WS, I, S, T or A) and say "it's worth X because the GM will win due to it."

It's also very much worth noting that the Baron would have Blessing (so a built-in ward), Finest Warhorses, +1M and Magical Attacks in addition to being much cheaper than the GM.
The only thing the GM would have going for him is +1I, +1 Save* and ItP to his unit. Is the GM worth ~45 pts more? Absolutely not, if anything he should probably be cheaper than the Baron, or at the very least be the same cost, +/- 5 pts or so.
(*Brets don't have access to Plate, but they do have access to a very good +1 save magic item if they want.)

Just for fun, another comparison:
A High Elf Prince with FPA and Shield on a Barded Elven Steed is 156 pts.
A Grand Master with FPA and Shield on a Barded Warhorse is 166 pts.

The GM has: +1T, ItP to him and his unit and Rallying Cry. (He also has Stubborn and Veteran, but those rules are only really useable when he is alone, and aren't something I'd be willing to spend many, if any points on, because he will rarely be alone.)

The Prince has: +1M, +1 WS, +1 Ld and Strike First.

+1 Ld (for a Ld10 General) easily trumps ItP and Rallying Cry (and once in a blue moon useable Stubborn/Veteran.)
+1 WS, M and Strike First easily trumps +1 T

Its funny tragic that the Prince is the cheaper option...

I'm honestly pretty sure that at some point, a Warhorse (or Barded Warhorse) was built into the base price of the GM and CM (it would explain their high basecost), and iirc in every earlier edition the Grand Master came with a mount by default.
Then they removed the mount, probably because they reworked how mounts are purchased in TOW, but forgot to lower the pts cost of the GM and CM. Would also explain why the GM has a plethora of rules on his profile (that he probably pays for...) like C.Charge, Swiftstride and F.Charge, when the very mounts he *must* buy for additional points all have the same rules...

It just doesn't make sense.

A GM should be in an army with lots of DGKs and Knights. However is that army good? Would it have state troops? How would the state troops perform?

I don't know about "should", a GM works in pretty much any army list.
Regardless of that however, he should be costed according to his own merits, not the merits of the armylist he's in.

More troops for the Emperor, this time an harquebusier unit. It's nice to see again on the field these nostalgic Perry sculpts form the early 2000s, the first wargaming models I ever had!

https://noigrimintothemaelstorm.blogspot.com/2024/04/the-eastgate-garrison.html

I was referring to the Bretonnian Men-at-arms. They are coming in at 51% which is the same (roughly) as the Bretonnian army. But there is nothing below them to balance out the things that are coming in at higher percentages so why is the Bretonnian win rate so low?

There are a couple possibilities. One is that armies that do not contain the men at arms and bowmen (such as the arcane journal lists) are coming in at a lower win rating. The other is that units that don't show up enough to make the 'cutoff' for unit stats, still have an impact and drag down the win rating of the armies that have them instead of other units. A third is that the time frames for the analysis are slightly different and sample sizes are still small.

Pegasus captain w/ full plate shield and lance is 87 points from memory. That's our cheapest fast chaff unit. Maybe that?

It's a very good buy for the points.

That's a good point. I saw "captain" and I was just assuming BSB. But you're right, that captain could be on a pegasus, or a griffon, and does not have to be a BSB. Maybe all the peg captains account for the difference?

I don't know... I have been taking fewer cannons overall, and haven't been taking any at 1500 points and lower. It's not just the cost of the cannons... the veteran state troops I take cost way more now too. Detachments work differently, and so does charging skirmishers, so overall my line units cost more and are less effective without adding character support. In 7th and 8th edition, I rarely used my full character allotment, now I struggle to make the characters I want fit. That impacts whether or not I can buy cannons too.

I don't know how many of these games are 1500 points versus 2000 points either. Maybe at 1500 points where Empire performs better, you definitely take a BSB, but maybe not a cannon, and that's what helps account for the win rate differential? 2 cannons plus an engineer is 300 points. That's hefty, and if I'm going for a cannon strategy I want some redundancy. It doesn't fit in 1500 points. So maybe we're seeing an artifact of point size?