This used to work just fine. We've been on Odoo9 and can set the price of variants on the pricelists. We're now migrating to Odoo16 and it looked like this is no longer possible and has become convoluted and not intuitive at all. Looking at the model, it would appear that there's some capability to price variants as the enumeration for the column product_pricelist_item.applied_on has the following values;

('3_global', "All Products"),
('2_product_category', "Product Category"),
('1_product', "Product"),
('0_product_variant', "Product Variant"),

This very much suggests that the developers intended to allow pricing at multiple levels. Eventually, I found that it actually works well by setting the configuration as follows;

Configuration->Product Catalog: Variants = True
Configuration->Pricing: Pricelist = True, Advanced Price Rules

When you go to add a new item to a pricelist, you can select what you want to price as per the enumeration values above.