When is it right (and non-pretentious) to use the word "concomitant"?

Concomitant is a formal word. We would generally expect only those people who are regularly involved with high register language to be familiar and comfortable with it. Even such people may find it awkward or pretentious if they encounter the word in a non-formal situation.

It's helpful to understand the word's precise meaning and to use it when we want to communicate that precise meaning in a context where people are likely to expect high register language, and when no synonym has a meaning or feeling that better fits our communicative purpose in that context.

Sometimes we feel that someone is using a word like this mainly to demonstrate that they know it. We often experience this as pretentious or pompous.

The only way to improve our ability to make better decisions in contexts near the margins between formal and not-so-formal is through experience. Asking others for feedback or opinions can speed this process.

The core meaning of the word conveys a relationship in time:

concomitant
adj.
Occurring or existing concurrently; attendant: poverty and its concomitant social problems. See Synonyms at contemporary.

It can sometimes be useful to search dictionaries for the term and notice the types of sentences used as examples. However, dictionaries often don't serve this purpose well (as the example in your question suggests). And perhaps lexicographers are not so concerned with helping us decide how, where, and when to best use a word as they are with trying to render clear its meanings.

(It seems to me that the idea in the example sentence would better be communicated something like She loved travel, even with all its pressures and worries. Or, She loved travel in all its aspects, including the worries.) I don't get a feeling that anything about the sentence besides the word is formal in tone, or that it belongs in a formal text.

The first example here represents to me an effective and appropriate use of the word in a sentence that might be used to communicate clinical or research information among medical scholars or professionals. The third example there, in contrast, strikes me as a poor use of the word. Entertainment Weekly is not a formal publication, and we can, I think, replace the word with usual to achieve a cleaner, more straightforward message without losing any significant value. My personal reaction is to think that the writer is deliberately trying to impress, and I'd tend to stop reading to speculate on the writer's insecurity and perhaps to roll my eyes.

We can often get more detailed and useful information on use in context from language corpora.