I love westerns and as they are few and far between I try to check out every one that comes across here. I'm at work so I don't know how far I'll get but I'll leave my thoughts.
The logline needs work. The logline is not an advertising slogan for the movie, it's a brief one or at max two sentence summary of the story. This logline as is tells me zero about your characters, setting or story.
You say there are 3 small children in the wagon but only mention two by name.
"They are not far apart in age." Cut this.
"Many hearts and hands are looking after his every need to quell the fever with cool towels and medicine." You've had a few descriptions like this already and it just reads awkwardly in a script. You are too general and need to be more specific on what is happening. For example, who's hearts and hands tend to his needs? How does a heart help? How would you even show that? Did you mean they were worried? Why not just say the Brumley women tend to Luke in a busy flurry of hands cool towels and medicine?
Page 2. If the scene is in the Brumley kitchen then include that info in your slug, not your action.
Also, lots of ing verbs. Some people don't mind them, others will. Instead of Luke feeling better just write he feels better. The women aren't preparing food in the kitchen, the women just prepare. Scripts takes place in the present and ing infers past actions. And reading on these are everywhere.
Slight concern. A lot of the dialogue is very on the nose. It's almost exposition the way they speak and nothing else. Your characters need more personality. Also show, don't tell.
Interesting. You jump 3 years from 1858 to 1862 on page 7. My question then is what's the purpose of the first 7 pages if your story starts in 1862? I understand westerns can be a slow roll but there's nothing that happens in the intro that couldn't be cut down and used right away in 1862. All I really know from 1858 is that Wash's fam is new to Texas, he was just married and it's dangerous.
Two separate coffee drinking scenes within two pages? is there anything else to do in the west?