The language inside doesn't matter. We're comparing the actions. One is using the Healer's kit as-is. The other is in a feature and that feature does not explicitly say you're using the "Use an Item" action, therefore you're not.Two separate sources using the same exact language, with a class ability about that very language, which changes the action to a bonus action. What is the in-game justification for it working with one but not the other? What is even your argument that this makes sense, if it's not "I don't like this combination"? Because if it's just that let's talk about the costs and benefits of taking a feat and taking this subclass to get this kind of result rather than this silly debate about how two identical sentences should be treated differently for apparently...reasons?
Healing someone and stabilizing them are different tasks, even with mundane equipment. It's impressive someone can bring someone back from unconscious to conscious within 6 seconds, let alone being able to do it before or after firing a bow in that same 6 second round. But that's like quickly administering a bandage or something.
On the other hand, I view healing them a significant amount akin to a quick surgery. You're pulling out complex equipment that requires full attention to apply, rather than just alcohol and bandages or a medicinal herb.
I don't think it would break anything for it to be ruled that they can do it but I don't want people getting up-in-arms about a DM interpreting it this way, because this seems to be how it was meant to be.