Upgrade in Dashboard tells me I'm already at the latest version (5.5.1) and doesn't do anything.
I also manually downloaded the 5.5.2 file package and copied it over the top of my existing installation. I used the /tools/required/upgrade.php tool per the upgrade instructions in the downloaded ZIP file. It too tells me that I'm already at the latest version and asks if I want to run the script again anyway. After running it says I'm upgrade to 5.5.1.
Sounds like the new core isn't in the right spot. Try extracting the zip archive into the updates/ folder at the root of your site. The path should look like:
(your site root)/updates/concrete5.5.2
Then go to Dashboard > System & Settings > Backup & Restore.
The URL method is really only necessary when your current core is somehow broken / not working and you can't get to the dashboard.
I took a peek at the upgrade instructions in the INSTALL textfile and while they *should* technically still work, you're right that the last time they got much attention was likely before the whole dashboard upgrade feature was introduced. We'll add revising those to our to-do list for next time.
(your site root)/updates/concrete5.5.2
Then go to Dashboard > System & Settings > Backup & Restore.
The URL method is really only necessary when your current core is somehow broken / not working and you can't get to the dashboard.