In a way, yes. You see a lot of explanations about topic vs subject, and emphasis on the left vs emphasis on the right, but here’s the way that made the most sense to me:
は vs が is, among other things, about where the information you want to give is.
With は, you’re basically saying "to the left of this is what we’re talking about, to the right of this is the information I want to give you about that. With が you’re essentially flipping that around.
So when you say この仕事をするのは私だ, you’re saying that この仕事をするの is the context for the information you want to give me. The actually relevant bit of information is 私だ. In a wordy sort of translation, it’s a bit like “On the topic of the one doing this job, the thing I want to tell you is it’s me”
Similarly when you say 私はこの仕事をする the relevant or new information is not that you’re talking about yourself. The relevant information is what you’re telling people about yourself, namely: you’re doing this job (as opposed to another job, or no job at all, or you saying you’re wearing green pants).
If you were to say 私がこの仕事をする, however, that means you’re giving people the information that it’s you doing this job, and not someone else. この仕事をする is the context for the information you’re giving, essentially, that’s not the “variable” part of the information you’re giving - you’re giving some information about doing this job, and that information is you’re the one doing it.
And sure, emphasis is one way of describing that, but emphasis is also a bit of an ambiguous term and doesn’t really clarify this - at least, not to me.
Does that help?
(Also keep in mind, I’m not fluent - there may be massive mistakes or misconceptions in there)