Personally I split it on 3 parts. Topic, contrast, implication.
Technically any topic has some degree of contrast, but not always this contrast is used for implication. Fundamentally when we communicate, we want to tell new information, but if only new information is provided, it's not clear about what we talk. It's like to say "ate the fish". Who ate? No idea, could be "I", could be "He", could be someone else. There are many potential candidates in such situation, so when we communicate, we have to provide some information about the topic, about what or who our sentence is to make it clear. This kind of disambiguation happens all the time, but there is a difference between providing overall topic, and replacing other particles that are fine by itself with は. The former is required, but the latter is optional and intentional. And this is why people typically talk about 2 functions of は particle. If person deliberately tries to disambiguate something that is fine by itself, it sounds like person tries to convey with it something else.
Such way if we ask if it's possible to use something as required information in a neutral way and at the same time as intentional implication, in my opinion it's not. Because either person tries to imply something or not. At least their intention should be only in one place.
Similarly topic doesn't have to be explicitly stated with は. In fact, there are 4 versions of it. Besides explicit は it can be context-topic, for example, when it's either previously mentioned or obvious. It can be predicate, for example, if people are about to leave a restaurant, it's time to pay. If in such situation we say 私が払う (I will pay), people naturally interpret 払う as a topic. Such usage is called exhaustive-が, but in my opinion it's rather caused by predicate topic. And finally when we talk about events, even occurrence itself can serve as a topic. Most common example is probably news, people want to know what is going on around us, but I like to give an example like "be careful, the brick is falling". If you think about such sentence, then information we want to deliver isn't about what the brick did today, but about situation itself and that people should react accordingly to it. Occurrence topic is common with event that are either useful to know or people expect some reaction to it. When we say a sentence like "John arrived", sometimes we mean "I know you were waiting for him, go meet him", and we talk not about him and his actions, but occurrence that let it happen.
So you are right in thinking that there must be some other topic. Similarly to how 田中さん exist in C sentence, there must be a person doing eating in A and B. Even if person doesn't try to imply anything, sentence like 猫はない sounds like person has some other animal. This is probably one of the main reasons why people can drop は completely and use zero particle.