Is it a better payout to separate the copper ingots from desktop heatsinks as well as the tubing connected to laptop heatsinks instead of selling it as a whole?

Absolutely, if your up to the effort.
Remove the steel mounts, then separate copper from Al.

Make sure to scratch test all copper to make sure it’s not plated aluminium.

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That round one with the copper in the center. If you have an arbor press you're golden, if not, a hammer and a rod to punch the copper out works too. Or if you have a way to cut the aluminum that's another option; but that makes aluminum shavings so I don't do that.

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Here to learn more so I can recycle more
My grades are my own opinion and not an official grade from Boardsort

Rich, this type of scrap item is a prime example of a "net-zero" profit item, that is, if you plan to pay to ship it.
Not to dissuade you or any other user from patronizing Boardsort, as they are honest and fair, but you are way better off cleaning this up and taking it to a yard that is local to you.
If you have no yard, then mail away. Keep in mind, that at current prices, and lets say you plan to use a large flat rate box or something similar, you need to jam 40 or so pounds of these in there just to break even on shipping charges. This is a hypothetical.
Most yards will give you on the order of $1.25 or so, maybe more maybe a little less, as a "radiator".
Pro Tip: jam a box with some higher paying items, think RAM or Logic Boards, then pepper these inside as decoration.

At the moment copper for boardsort hovers mid $2 range. Meaning 10ish pounds in a large flat rate box to break even.
But local rates could be higher or lower.

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Hello All.

Happy to join the talks, guys.

I prefer to separate Cu/Al heatsinks.
As far as it is pure Al it goes as highest grade Al at selected scrapyards.
Heatsinks could be reused in many applications. I sort them by weight/dimensions and put similar items in the separate boxes and stick a label with item weight/sizes.

Hope this helped,
Regards