Looking at the UA of both Sweden and France, it's clear that Authority seems like the ideal ancient era policy tree. For Sweden, it's an absolute must with their synergy with Great Generals. With France, I guess you can pick other routes and still make it work if you're pretty good.
What's the strength of Authority? They are meant to give you a significant edge in early game wars. While bullying CS and hunting barbarian camps are vital, they cannot compare to your defeating your neighbors. Capturing enemy cities means you grow stronger and your enemies grow weaker. That's how you snowball.
Now, when do we war? Generally, it's either the early game when you are more or less on par in techs with your enemies or when your UU comes online. If you are Sweden or France and you wait until you get your Tercio UU, then I believe that's a big mistake. You'd be behind (degree depends on your difficulty) in techs and, on Standard speed, the window of opportunity for you to really take advantage of your UU is going to be smaller. It's doable but it's going to be hard for sure.
Why am I bringing this up? I guess I want to figure out the purpose of the UB. Public School is available late mid-game but I guess it's usually late game for higher difficulty since the game is entering a stage where it will be ending soon with a snowballing player or AI gunning for that finish line. I'd like your thoughts on this because I can only share what little experience I have with that era.
First, we'll look at the French UB. The Grande École gives 1

Production and 1

Science for every 2

Citizens in the city. I think it's fair to say that cities are reasonably large at that point and, if the player is doing everything correctly, then the capital can have a very high population. In addition to more yields from specialist which is quite possible at that stage unless you halted growth a lot, you also get more yields from the building itself (or am I misreading the +3

Production, +5

Science and +1

Culture. Depending on whether or not you need more

Production or

Science, you can change what specialists you want to prioritize. That is certainly something you don't enjoy with the Swedish Public School UB. The Grande École is (always?) a must since France does well with Artistry and Louvre is that Wonder you should get along with the ability to get Archaeologists.
Secondly, we'll look at the Swedish UB. The Skola requires less

Production to build and less Maintenance. The Maintenance shouldn't matter at that point unless you're really starving for gold. At that point, I think you have bigger problems. Being able to get the UB faster is definitely a big plus regardless of the speeds. You get 1

Science for every 1

Citizens in the city. That's double the

Science of France if you have same sized cities. It also gives 1

Culture for every 3

Citizens which, while nice, isn't that impressive. There was a reason why people stopped taking Inspiration when it was nerfed to that ratio. Inspiration also came earlier when

Culture played a larger role so you can imagine the impact of the Culture you can get by the time this UB is available. The Skola does give 2

Scientist Specialist slots compared to the 1 from the Grande École so that's more potential

science if you want to work it. One issue I see here is how Public School requires Scientific Theory while Fusiliers comes from Rifling. Do you prioritize wars or your UB? That's never an easy choice if you aren't ahead and, if you are ahead, then you probably already snowballed at that point.
Can anyone confirm is the Grande École reduces more Illiteracy than Skola? Ones say Greatly reduces while the other is just reduces. That can make a big difference too.
Let's look at the French UI now. Chateau is certainly one of the stronger UI in my opinion as those yields are quite strong. The requirements for where you can build it can be rather dependent on the map but that's part of the fun. Chivalry will be prioritized if you went Authority anyways so that's perfect. The extra defense bonus can help you whittle down enemy forces before making a push yourself.
How about the other Swedish UB? This UB, in my opinion, seems very buffed. Maybe it's buffed due to how alright the Skola is. Baths are great building with a terrain requirement that isn't possible. The Bastu ignores that which in itself makes it really good. The fact that it promotes you to grow your cities more makes it work very well with Skola as you'll have enormous

Science output at that point. I like the way they synergizes well together but they don't make Sweden OP. Why? Sweden should be utilizing its UA early game so, by the time Metal Casting is available, you should be in a decent position.
What are my thoughts of these two UB? Skola is in an awkward place in the tech tree and requires you to make a difficult decision if you wish to war more or not. I do wonder if it's worth it to go Skola when you can instead strengthen your military and just conquer your neighbors. Meanwhile, Grande École works really well for France who doesn't necessarily needs a Domination Victory to win. Going for CV is certainly viable with the French kit. In addition to some powerful yields like

Production, you also get a lot more versatility with the Grande École. I think a slight nerf might be needed for Grande École but I have to see it in action. Just looking at the numbers on paper, Grande École is, in my opinion, at least 2 times better than Skola. Looking at the whole kit, I'd have to say that France has more versatility when it comes to DV and CV while Sweden seems to have to make a detour if this civ wants to bring the full potential of its kit.
These are just my two cents though. Correct me if you think I didn't make a valid point. I like to hear other people's thoughts on this.