.....and to answer your question, the rules of golf are paramount. Other than Stableford, there is no occasion in a singles stroke play event where a player who does not hole out isn't DQ'ed, thereby making his round unacceptable for posting.
You said: Now, on to Section 4-2. This again uses most likely score, but brings handicap into the picture. Par plus handicap shots is applicable. There are two parts to this rule, one is holes not played, one is holes not played IAW the rules. Now, for the first part, assuming you are not playing match play, in which case the only way to not play a hole is for your opponent to concede you a 1 on the tee, you are playing stroke play. Well, Rule 3-2 says "If a competitor fails to hole out at any hole and does not correct his mistake before he makes a stroke on the next teeing ground or, in the case of the last hole of the round, before he leaves the putting green, he is disqualified." So, simply put, any round in which the player does not hole out, other than in match play, is an invalid round, therefore ineligible for handicapping, or so says the rules.
That's not the case when playing a fourball, match or stroke play, as the rules for that game allow picking up before holing out.