Your point about putting a weight at an asymetrical point on the line and the line changing agnle between the weight and the two attach points is correct. I've no argument with that.
But there's nothing in the rule book that I know of that says that catenary is by definition symetrical. Catenary is the sag in the line, period. What I take issue with is your apparent claim that the portion of the line that is at a steeper angle is not contributing to the overall catenary (sag). This is not correct. Every inch of line between the two suspension points is having an effect on the shape of the catenary, whether you hang a weight on the line or not, and where you hang it if you do.
If you hang a weight on a rope rode down near the anchor, sure, it will reduce the angle of pull on the anchor. Depending on the amount of pull on the rode and the weight of the weight, there is an optimal place to put it to reduce the angle of pull on the anchor as much as possible. Go either side of that point and the angle of pull on the anchor shank will begin to increase again. I have no idea how to arrive at that point.
Because an all-chain rode is so much heavier than a rope rode, it helps eliminate the need for a separate weight to be hung from the rode. Not that a wieght hung from a chain rode won't help lower the angle of pull; weight's weight so making the rode heavier in the middle or wherever will encourage it not to be pulled up higher by the boat pulling on it.
But the point at which the angle of pull becomes high enough to be concerned about will be arrived at a lot later with an all-chain rode than with a rope rode with no supplimentary weight on it. Which is my sole reason for prefering an all-chain rode over a rope rode or mostly rope rode for our boat.
This catenary thing not some new secret. It's so obvious even my dog knows about it.
I''m simply saying that the entire suspended length of a rode, rope or chain, weight hung on the rode or now weight, and regardless of where that weight is hung, has an effect on the overal catenary (sag, curve, pick your term) of the rode. Including the part that has a steeper angle than the rest of it.
It's at that angle for a reason and in the overall scheme of things, that reason is important to the rest of the equation.