I've been attending Gen Con for 10 years, and for most of that time I did not buy event tickets until July. Many events are sold out, but many others are not sold out!
To choose events, I recommend downloading the Excel file from http://www.gencon.com/experience/find . Then you can filter out sold-out events.
I believe the "Badge Registration is now OPEN!" and "Event Registration is now OPEN!" signs on the front page previously gave the dates when they would open. There's also a news feed on the left side with this link to sign up for their weekly newsletter.
Unfortunately, none of that appears on the mobile version. I'm guessing that's how you never saw it.
Here is how the ticketing system works:
You need a Gen Con badge to get into some areas and to buy event tickets. If you want to spend all day shopping and playing demos of games in the vendor hall, the badge is all you need.
Most events have event-specific tickets.
A tabletop game could easily be limited to 4 people, so it wouldn't make much sense to let 100+ people form a line to get in. Gen Con has you buy tickets in advance, and you cannot have tickets to two events with overlapping times.
Gen Con takes a fee for the space, and beyond that prices correlate to the host's cost to run the event. Not all events cost money, but they still have a $0 ticket that guarantees you a spot where attendance is limited. It's free to attend seminars and movies, to demo some new games or playtest works-in-progress.
You buy tickets online and pick them up at Will Call at any time during the con.
If you want to join an event for which you don't have a ticket, and it isn't sold out, you can spend "generic" tickets of equal cost.