There is a concept called topic-prominent in Linguistics field. I think this theory has the key to understand the difference.
In many cases, you can translate “Nは” to “As for N”, “In case of N”, or “Now I’m going to talk about N”. It’s like setting a topic.
When you introduce yourself you should use “は” because you have to set the topic of you.
As for me, I’m from Tokyo.
If you use “が” in the former example, it sounds like you aren’t describing about yourself (probably because of topic-less).
I (not him) am from Tokyo.
This sounds natural as long as answering for a question like “Who is from Tokyo?” or “Mike is from Tokyo, isn’t he?” This is not to introduce “myself” but to clarify a contrast, difference or misunderstanding.
One important point here. To understand the difference between は and が, we have to clarify the difference between “introducing” and “describing (from 3rd party’s view)”.
(1) (We’re going to talk about Mike,) he studies Japanese.
(2) Mike studies English.
(1) sounds like introducing Mike. So if you were asked “What did you see in the library?”, マイクは日本語を勉強していた is bad because now the topic is “what you saw” and not “Mike”. わたしが見たのはマイクが勉強しているところです is perfect instead.
(2) has no topic. It sounds like describing Mike from 3rd party’s view. So if you were asked “What did Mike do yesterday?”, マイクが日本語を勉強していた is bad because it sounds like irrelevant story from “what Mike did”. Obviously started to talk about “Mike” so マイクは日本語を勉強していた is good.
(3) It’s getting popular.
(4) As for popularity it’s getting better. (But sales is not good…)
(3) is very common. we usually talk about “popular things”. When you want to set “healthy food” as a topic, you can say ヘルシーな食べ物は人気がある. You can also say ヘルシーな食べ物が人気がある it sounds more like describing healthy food fairly.
On the other hand (4) is not common because we usually don’t specifically refer to “the popularity of something”. So it has strong meaning like “As for popularity” or “In case of popularity”. Thus in this case (4) leads to contrasting effect against the example of "わたしが東京出身です".
Conclusion. In your example I don’t think ヘルシーな食べ物は人気が出てきた is bad. Because they have already talked about healthy food like fishes and tofu. So it’s natural to set “healthy food” as next topic by using “は”.
However ヘルシーな食べ物は人気は出てきた this is probably also possible but it sounds confusing because topic is multiple. If you want to emphasize the contrast against ‘sales’ for example, you can say ヘルシーな食べ物の人気は出てきた.