Etsy can be linked just like other bank or credit card accounts. You can check the detailed steps here: Connect to QuickBooks Self-Employed.
Also, I'd like to know what exactly happens when you connect it. If you're having difficulties downloading the transactions, you'll want to refresh it. Go to the Gear icon and select Refresh all.
If the same thing happens, please perform these additional steps:
Use an incognito window and link, refresh, or check the banking from there.
Google Chrome: Ctrl + Shift + N
Safari 11 or newer: ⌘ + Shift + N
Mozilla Firefox: Ctrl + Shift + P
Microsoft Edge: Ctrl + Shift + P
Clear the cache. This causes issues in QuickBooks Self-Employed.