Would it negatively affect the recipe to transfer all of the ingredients into a slow cooker, once the short ribs have been browned and the onions...

Agree with the above answer; also I might leave it a little longer than five hours, maybe six, depending on how powerful your slow cooker is. I've found that with my fancy All-Clad slow cooker, a beef stew that says to cook on high for seven hours can do the same on low with good results, but with an inexpensive slow cooker you might be safer on high, or a dual setting where it cooks two hours on high and then 3 or 4 on low. I haven't looked at the recipe, but as with all slow-cooker cooking, no peeking and make sure you have enough liquid.

Not my recipe but I see no reason why it wouldn't work. It might even improve it.