In many poker variations (5-card draw, seven-card stud, etc.), players' hands are disjoint, so any card which is present in one player's hand will be absent in everyone else's. In such games, there would no need to consider kickers as a tie-breaker for hands showing three-of-a-kind or four-of-a-kind, so there would be no need for the rules of those variations to mention them.
The need to use the kicker as a tie-breaker between hands showing two-pair, or to use anything past the top card as a tie-breaker for two flushes, would be much less common in disjoint-hand games than in community-card games. It's plausible that poker was widely played without those tie-breakers in days before community-card games became popular; provided all players agree what rules are being used, the game with those tie breakers would be essentially the same as the game without.
The notion that only the top card is used as a tie-breaker with flushes may be reasonable in disjoint-hand games (if all players agree to it), but is would adversely change games with community cards. Further, it's probably more practical to use the same hand rankings in all games than to use different rules for the different poker variations.