Kanji have exceptional readings, in addition to their on and kun readings.
You won’t find any part of きょう listed as a reading for 今 or 日, even though putting them together makes that reading.
That’s just a fact of kanji.
Now, お早う is a bit different in that it’s an archaic conjugation that only exists in modern Japanese in that greeting (as well as some other set phrases), but the principle still applies. You will find ways of reading kanji that aren’t listed for that kanji’s on or kun.
And as MasterSenSenpai said, there’s no reason to be mad that you can’t “properly study” お早う because you’re extremely unlikely to ever see it.