Yes, "ought" is the past tense of "owe". At some point, the second alternative spelling "owed" was introduced to better separate the two meanings (literal and figurative), but it's still the same word; a similar thing happened with "flower" and "flour", those used to be interchangeable spellings of the same word but then somebody decided that the two meanings of that word should be separated and given specific spellings.
And the construct "you owe it to <person> to <verb>" still exists even today but is not nearly as popular as "you should <verb>" precisely because it has to state to whom exactly your owe the duty; with "should" it sounds like an impersonal, quasi-objective statement of fact which suits the manipulative uses much better.
I have occasionally used the construct “you owe it to yourself to X”. I think it works well at conveying the sentiment that the person in question may be missing out on something if they don’t do X.
“You should” has a much more generic and less persuasive sentiment. “Why should I?” is a common and easy response which now leaves the suggester having to defend their suggestion to a skeptical audience.
Good point about "should" - it's also a word that has lost its original meaning. Shall, should, will and would used to have different, more nuanced meanings comprared to how we tend to use them today.
The only place today I see "shall" used correctly where most would say "should" or "will," is in legal documents and signage.