These recipes seem to be a misunderstanding how stuffed pepper recipes typically work. At least when we are looking at the Balkan tradition, where the dish originated - this answer focuses on it only. If there is by now a changed form in US recipes, it is not included in my use of the word "traditional".
Traditionally, stuffed peppers are made with sivriyas or other peppers with at least one small dimension, not the monstrous spherical things available in Western supermarkets nowadays. The rice is prepared in the pilaf way - it is fried in oil, then it can be slightly parcooked or just be used at that stage. It is mixed with the other filling ingredients (much more rice than meat) and stuffed into the peppers after they have been punctured in multiple places with a fork. They are placed in the baking dish and the liquid is added.
When you do it that way, the rice cookes in the oven. The liquid you mention is needed to cook the rice. The holes are needed for the liquid to penetrate into the peppers, the small dimensions are needed so that the liquid reaches even the middle, and the "small pieces of meat embedded in rice" texture is again needed for the liquid to get throughout the rice.
When the peppers are ready, they will still have some liquid left over in the baking dish. But after an hour or so, the rice will soak up the rest of the liquid, and you'll only serve the dish after it has rested.
Beside that, it is pretty common to eat the prepared peppers with an additional liquid - mostly yogurt, but I have also seen tomato sauces, but never something as thin as broth. These sauces are added at serving time, traditionally. For eating, you don't spoon the sauce, you break off parts of the stuffed pepper and mix the rice into the sauce before scooping it up. It is possible that the people who made the recipes in your link are using the liquid in that way.
It is also possible that they are just taking the peppers out of the sauce before serving and discarding it. Note that one of the recipes mentions as a last step "drizzle with a spoon of pan juices" - this suggests that you are only using a small part of it.