This contradicts what you taught in the tips&tricks, doesn't it? You explained that "rat" is only used to indicate the adjective and not to exaggerate. I would expect "we are sorry" to be correct.
rất means very/very much. rất xin lỗi = very sorry. Chúng tôi xin lỗi would be "we are sorry".
Good to know. The confusing part is that the tips present it kind of like the Mandarin 很, i.e. it's just there for simple NOUN is ADJ sentences, not really meaning "very" so much as just being a part of the NOUN is ADJ construction.
I thought the rất was necessary...?
Nope. If you just want to say I'm sorry then you just say "xin lỗi". You can even through in a particle like "nhe/nha" as in "xin lỗi nha" to soften the tone of the apology.