Mar Ukva said: Regarding [waiting between meat and milk], I am like vinegar derived from wine [i.e. much inferior], compared to my father. My father, when he ate meat, would not eat cheese until the same time the next day. But I do not eat [cheese] in the current meal, but do eat in the next meal. (Chullin 105a)
The halacha has been decided like Mar Ukva, that between meat and milk we wait from "one meal to the next", defined as either 1, 3, 5, or 6 hours. But I think the father's approach of waiting 24 hours is more logical than Mar Ukva's, even if nowadays our practice is otherwise.
The concept of waiting 24 hours is well known in the laws of kashrut. According to the halacha of ben yomo, a utensil which has been used with meat or milk acquires a milk or meat status for a period of 24 hours after use. After that, any food residue on the utensil is considered to have decayed to the point of inedibility and may then be eaten with the opposite "gender" (according to the Torah, but rabbinic law is generally stricter).
This principle might logically seem to apply not only to utensils, but also to one's mouth and stomach which come in contact with meat. Surely once you eat something, a little bit of it remains floating around your stomach or stuck between your teeth. If this kind of residue is enough to prohibit using a utensil with milk for the next 24 hours, why should your digestive system be any different? Based on the principle of "ben yomo", we should wait 24 hours between eating meat and milk (and perhaps between milk and meat, though even Mar Ukva's father might not have done that). In contrast, our custom of waiting 1 to 6 hours has no such theoretical basis and seems to be pretty much arbitrary.
So the thought expressed by the 24-hour opinion is not just "stricter is better", but also "more logical is better". And yet, we ignore both the safety of stringency and the attraction of logic in following our own custom. Sometimes our customs are annoyingly particular, but other times they provide us with great freedom. By following even the customs which take extra effort, we gain the legitimacy to rely on custom for leniencies as well.