As @SergioDavid requested, I tested my Spotify account in a colleague's computer, and it presented the same problem: the web player doesn't work. However, when using my colleague's account the web player worked perfectly.
So it seems it could be related to the account. As explained in the original post, the issue began manifesting around the time I purchased a Premium subscription. Around that time, I also changed the password and the email associated to the account.
I tried changing the password back to the original one and the web player still fails. I haven't tried changing the email back. I'll try to test that tonight.
Hey @user-removed! Thanks for your post!
It sounds like this is related to your account. Spotify Support can take a closer look. You can reach out to them here or on Facebook/Twitter on @SpotifyCares. Feel free to link to the thread you made. Let me know if there's anything else.