I find it's helped for a few reasons, but the big one is just that it's easier to keep the colours in my head - I'm not thinking in terms of "tertiary-alt-3" or something, I'm looking at a design and going "that's green, which green? This green". I have more discussions talking about colours in terms of the standard English colour names than I do in terms of those colours' purposes in the design system, so the colour names seem more practical.
That said, I agree that spending some time switching the colours to whatever design system you're using us very useful. Another trick I've found is starting off a project just using raw CSS values (the `p-[8px]` syntax), and then when the basic designs are done based on whatever the designer has sent, getting a list of all the custom classes you've got and using this to define what the standard units are. This is helpful when the design guide isn't specific on these details, and the designer is being more freeform.