I don't live in Japan so I won't claim to know a lot about their culture, but in Japan and especially in the workplace you're expected to be very formal and polite, even more so if it's your first day. And Rei just...not only doesn't greet anyone, but even talks back to Elena (effectively his senior in the company), which is kind of unspeakably rude and disrespectful by Japanese standards, lol. (I guess manager is a higher position than idol, but that doesn't excuse his total lack of manners to people he meets for the first time)And for me, out of that two, it actually was Elena that was extremely rude, and not Rei. Plus, it looked like they saw and treated him like a free slave. I hardly can imagine Anezaki going to bring a water to either of the girls because they just said they wanted it. But maybe it's just me, and me not knowing the rules of show business...
From me his behaviour was pretty natural, since he's a higher position than an idol, he doesn't really have to greet them, and he came there to work, not to greet people. So his "let's go" to Eri looked pretty natural and normal to me. But then again, I also do not live in Japan, so maybe indeed out there it sounded rude. However I myself do work on a movies studio, and I'm daily dealing with actors, actresses, directors, producers, screenwriters, bg actors, and people that think they're bigger stars than they really are, and real stars that for a change usually are pretty humble. And greeting someone is pretty rare, and happens only after working for a longer while. Enough that people will recognize you. Newcomers are never greeted.