When the installation finished, you were prompted to run Photoshop and so you did.
Re-install the extension, except; do not run Photoshop when prompted. Rather run Photoshop once the installation is complete.
When the installation is complete and I'm prompted to run Photoshop, I see the fontea extension in Windows > Extension. Although, if I don't run Photoshop immediately after installation and run Photoshop thereafter the installation is complete; I don't see the fontea extension in the Windows > Extension list ?
I'm sorry - I wasn't prompted to run Photoshop at all. I did so of my own accord, to see if the installation worked.
From what you're describing, that last sentence appears to me to possibly be an issue with Photoshop, itself. You say it shows up at first, then disappears. Are you running any other extensions? If so, I'd suggest there may be a conflict, and so turning them all off (or removing them from the folder, temporarily) will answer whether there's a conflict. It's possible something else needs to be upgraded.
Not long ago, Photoshop had a limitation on Filters - if you had more than 6 installed, only the first alphabetical 6 would show up. Perhaps one of your other extensions is making Photoshop act in the same way for this issue.