English is Primary

Interesting question... my answer is, probably (?) Depends on the scenario, and who is your market

In an english-speaking class within an english-speaking country, a local trainer might not need to say that they can only speak in English however, a migrant such as myself, may need to state (with proof) that I am fluent in English so i can land the job.

On the other hand, in a diverse class, it could be a competitive advantage to be multilingual as compared to those who only speak english exclusively, as you may be able to communicate better with the diverse students using different language/s. Having said, if the client expects a multilingual trainer, yet you can only speak english exclusively, then you may need to explicitly state that so that both you and the client are aligned with the constraint.

I may not promote it in socmed or in my bio, but it could be in my letter to the client or state it during the interview where applicable.

I hope this helps!