I have spoken with Birna and purchased the Coral Dragon Claw from her, but the quest "Ashore in a Sea of Ghosts" doesn't start. I tried loading a previous save wherein I had not yet gotten the quest from Jarl Korir to retrieve the Helm of Winterhold from Yngol Barrow; even after doing this, I am still unable to receive the quest.

What is the problem? How can I fix this and start the quest?

You don't get a quest when buying the claw from her.

How do I get the quest "Ashore in a Sea of Ghosts" then?

There was a button labeled "activate" that I could not manipulate and a gate that I could not open, which I assume is opened by the button. When I encountered this obstacle, I looked up Yngol Barrow on the net and learned about "Ashore in a Sea of Ghosts" and the Helm of Yngol. The helm wasn't on the throne; the throne was empty after the Draugr Scourge Lord vacated it.

How do I start this quest and how do I get the helm to spawn?

UESP doesn't even list a quest of this name, it has it as Yngol's Barrow. I wonder where the wiki got the name from. Anyhow, it doesn't appear in your journal, so stop worrying about it.

You should have fought Yngol's Shade, not a random draugr. Did you check if it shows up in the save where you don't have the quest for Korir?

Will try that when I play tomorrow. Thinking of going in there with the Coral Dragon Claw and without the quest from Korir, then returning with the quest from Korir to loot the Helm of Winterhold from the chest.

Yeah, there's a bug where, if you get sent to Yngol Barrow by some radiant quest - such as getting the Helm of Winterhold - the Helm of Yngol will not be there. Happened to me.

If you've already got thr helm of Winterhold for Korir, wait 30 days for Yngol Barrow to reset.