You're on the right track, but not 100% on it imo. For one, I don't think you strictly need to beeline Currency or Apprenticeship against all other priorities. Currency is great, but you don't really need it until you start building districts, which for me doesn't come until after building up my army and conquering a bit. Still an early priority, but there are plenty of techs besides mining you might grab along the way.
As for Apprenticeship, IZs are a lot less vital since the patch. Like Currency, you still want it, but not at the expense of every other tech, and in this case it's not even my top pick of the era. That would be Machinery, which unlocks crossbows and lumber mills. Those are great for many reasons, but here's two: for one, river lumber mills are some of the best tiles in the game, for another, they help boost production in cities that don't have hills.
As for districts, for me it's usually commercial + whatever my victory district is, eg theater for culture. Industrial and Entertainment as necessary to cover your empire, with a couple good campuses for any VC.
Another consideration whether or not to beeline Currency/Apprenticeship is access to Great Person points. I have found Great Merchants and Great Engineers to be the strongest Great People in the game.
For Classical Great Merchants:
Zhang Qian provides an extra trade route capacity
Colaeus grants you a luxury. Or if you save him, he gives you an interface to easily view all the luxuries on the map.
Marcus Licinius Crassus gives you a small amount of gold but can indirectly save you a lot of gold by obtaining expensive city tiles for you. Alternatively, he is currently the only way in the game to actively obtain tiles that are greater than 3 tiles away from your city center.
For Medieval Great Engineers:
Isidore of Miletus provides production. He can either rush wonders (always useful), or the production can be converted into hammers for other things (like districts or projects) by using overflow. Build a wonder to within 1 turn of completion, then use Isidore (or Brunelleschi or Eiffel) to finish it and get overflow production.
Bi Sheng allows you to build an extra district. This ability is only okay. Sometimes it can be particularly good to ensure you get both a Commercial and Industrial district in your capital early. It is best used with a city that has the Oracle since you maximize your Great Person points. The Eureka to Printing isn't that great, but sometimes it can net you some envoys due to CS quests.
James of St George isn't so great. In many games, it's not worth it to build walls. Walls don't seem to have long-term value, especially once you get Civil Engineering. Not to mention you will be paying for maintenance all that time. The one advantage to James of St George is that since Great People can instantly teleport between cities, it means you don't have to build *any* walls. And then if a city does get attacked and you think you need walls, teleport James there and have him magically create walls. But he is definitely my least favorite of the Medieval Engineers.
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If you want multiple Classical Merchants or Medieval Engineers, you probably need to rush Currency or Apprenticeship to start building those districts early.
The problem isn't so much the competition. The AI tends to be slow in building Commercial and Industrial districts, though Kongo can give you a run when it comes to Great Merchants. The problem is if too many AI's reach the Classical or Medieval era, the new Great People will be from the following era.
