The difference between transitive verbs and intransitive verbs is in how verbs relate to their objects. It is true that 「が」is used with intransitive verbs, but the problem we're dealing with here is not about verb/object, it's about subject/verb.
Which is to say, one of the main functions of 「が」is also to mark the subject of a verb.
Since the first sentence that you cited is lacking context, let's focus on the excerpt from the story.
In the first sentence, we come to learn that, when the little girl left the house, she put on some shoes. Since 「履く」is a transitive verb, it requires an object,「靴」, which takes the 「を」particle. But now that 「靴」has been established as the object of 「履く」, it's unlikely that we will ever see 「を履く」again in this part of the story, which is why I said that this is not an issue of verb/object.
In the next two sentences, we learn that the shoes were essentially useless because they were too big for her feet. Why? Because, in the last sentence, we find out that they were shoes that her mother had worn long ago. As I said, 「靴」had already been established as the object of 「履く」, so 「靴」is omitted. (This is similar to English, where after the object has been identified, it is thereafter referred to as it, them, etc.)
But, we still needed to know who wore the shoes long ago. So now the author uses 「が」to mark the subject, her mother, as the one who used to wear them.
Applying this explanation to the first sentence you cited, Sakura thinks that Matsumoto will stop doing something, but that something would have been established in a previous sentence, so now it has been omitted, and 「が」is functioning to mark Matsumoto as the subject who is stopping/quitting.
So, in summary, when the object of a transitive verb has already been identified, it can usually be omitted. But if the subject who is acting on the object changes, 「が」becomes necessary to identify that subject and it is irrelevant whether the verb is transitive, because again, it is no longer a verb/object problem, but rather a subject/verb problem.