Why I am not seeing the coral dragon claw on the pedestal after the puzzle in Yngol Barrow?

Solution 1:

If you go to the barrow before going to see Birna, is should be in the barrow. However, if you go to Birna first and do not take the claw once offered [e.g. if you missed it], it will disappear from the barrow.

In order to obtain it from Birna in the first place, you have to ask her if she has ever had a bad deal in her business:

If you ask her if she has ever had a bad deal after asking about her business, she will explain how she came to possess a seemingly useless dragon claw.

https://en.uesp.net/wiki/Skyrim:Birna

If you didn't take the claw from Birna, after it was offered it will not be on the pedestal either.

Once you've offered to buy the coral dragon claw from Birna (even if you didn't actually do so), the one on the pedestal in Yngol Barrow will disappear.

https://en.uesp.net/wiki/Skyrim:Yngol_Barrow

It seems to me that what most likely happened, if it's not in the barrow, is that somehow you missed the chance to purchase it from Birna when you spoke to her. That means that you either have to go back to an earlier save or forego the quest.

You could always go back to Birna and go through all dialogue options with her and see if you missed the offer, of course looking for the option about the bad deal.