I agree that CATCHING the accident is a good way to communicate your expectations, "no, not here, outside!" BUT, puppies are SO fast, and sometimes catching them isn't easy, if they aren't right there in your line of sight, even if they're across the room, it's hard to get to them in time.
Also, correcting inside accidents can lead to problems. Puppies/dogs don't think the way humans think. So, if they are "correcting" for peeing inside, they can sometimes think peeing in general is bad, so they will work harder to hide it from you or develop a thing about not wanting to pee in front of you at all, even in the right place.
I agree puppies need to learn independence; that's super important. However, you can't catch accidents if you can't see them, and the more accidents a puppy has, the more of a habit peeing inside becomes, and that is a habit that can be hard to break.
My opinion is you have to compromise between having your pup right with you, and also giving them some alone time. For instance, we just kept our dogs in our sight when we were potty training, we didn't use a tether (didn't know about it yet!). BUT, when I needed to do laundry or cook or shower or if I needed to check my email, we'd put them in the ex-pen with a kong, or some toys. So, they did develop some alone time skills.
I'd suggest only giving them added freedom when you are pretty sure they are empty. Otherwise, keep them in your sight, or contained when you can't watch them. Six months is when they really have complete control of their bladders, so accidents are possible before that, BUT, if you are on top of it, taking them out often, it doesn't mean you HAVE to have accidents. You just manage it, so that you get them out when needed.