What is the best way of creating a modular texture?

Hey there! So my problem is as follows: I want to be able to create a lot of characters. The features and textures of said character have to be modular eg. skin color, different eye textures, hair textures, different armors and so on and so forth.

So I used to do it by having a gameobject with lots of child objects, one for each body part. This would be about 10 - 20 game objects per character, and If I want to have alot of characters, this will add up quite quickly.

So instead I want to create a new texture based on all of these things. It reads the body data (eg. what head shape, eyes, mouth etc) and the equipment data to generate me one single character texture, which would cut down the amount of objects massively.

So I tried to create a new texture2D in code and calculating all the offsets by hand and this grew messy quickly. Another Idea I had was to use a character shader instead which reads all of the data.

Would the shader idea even be an improvement resource wise?

Is there a standardized way of doing something like this?