The evidence supports something freeing: there is no correct number. Wash frequency is a matter of scalp type, hair type, and preference — not a hygiene law with a universal value. So let's answer the pieces you actually asked.
Oily roots by day two: washing that often (or daily) is fine. The "scalp training" idea — that stretching washes teaches your glands to produce less oil — has no good evidence behind it; sebum production is driven by hormones and genetics, not by how long oil sits on your head. Stretch washes if you prefer the ritual or your hair likes it, but don't white-knuckle through greasy days expecting your glands to learn a lesson. They won't.
Dry ends: that's not a washing-frequency problem so much as a where-the-products-go problem. Shampoo is for the scalp — lather at the roots, let the runoff do the length. Conditioner is for the mid-lengths and ends, not the scalp. Add less heat styling and looser hair elastics, and "straw" usually improves within weeks. Your roots and ends have different needs; treat them as different territories and both get what they want.
One exception worth flagging: if your oily scalp comes with flakes or itch, that's often seborrheic dermatitis territory rather than simple oiliness, and a medicated shampoo in the rotation is the actual fix (https://dermnetnz.org/topics/seborrhoeic-dermatitis).
So: wash when your scalp feels like it needs it — for you, plausibly every day or two — condition the ends every time, and unsubscribe from anyone legislating your wash schedule.