Is it acceptable for supervisors to communicate only in Japanese?

Yes, using only Japanese is acceptable.

Foreign nationals coming to Japan under the Specified Skilled Worker, Technical Intern Training Program, or Employment for Skill Development have acquired a certain level of Japanese proficiency before starting work.
Therefore, basic work instructions can be given in Japanese.

[Tips for Smooth Instruction]
What matters more than language proficiency is conveying information clearly.
・Keep instructions short and specific
・Avoid overly long sentences
・Use photos or actual objects to explain
・Incorporate gestures and pointing for confirmation
・Use “Easy Japanese”

Example
× “Make sure you do this properly.”
○ “Please insert this part here and tighten the screw.”
Even a slight increase in specificity can greatly improve understanding.

[Key Points by Status of Residence]
・Specified Skilled Worker
Since these individuals have some work experience, the key is to organize and explain technical terms clearly.

・Technical Intern Training Program / Employment for Skill Development
Since this system is based on training, step-by-step explanations and repeated confirmation are particularly important.

・Engineer/Specialist in Humanities/International Services 
It is important to carefully share technical terms related to the job, internal company rules, and standards for reporting, consulting, and communicating.