Fortunately, I fixed the problem before I had to disassemble my laptop (again). For the sake of future frustrated Googlers with the same problem, here's what *I think* I did to solve it.
First, I manually adjusted the system date and time to bring it to today, and within a few minutes of the actual time. Then, I installed all pending system updates and let the computer shut down. When it restarted, I tried to sync the time again, and it worked.
I think the main cause of it was the incorrect time/date. When I tried to load web pages, I couldn't connect because my system had the wrong info (security feature of some sort I guess). I think this also prevented me from connecting to a time server, but when my system restarted with the right date, it could access the internet again. Of course, I could be completely wrong.