The current steps to access my profile are the following :
- Click on the menu
- Click my name which brings me to the Stack Exchange profile
- Look for the profile I want (sometimes even need to click see more to get to the one I'm looking for)
- Click on the almost unclickable link to finally access my profile page.
Per site profiles should be accessible directly on the website that you are visiting or by a quick button in the menu. Anything faster than this process above.