I don't read all emails, none except the important ones. I have Gmail and I just look at the subject line. I get a lot of emails trying to sell products, and its obvious by the title subject line... I don't have enough time to read emails, and I'd probably go crazy because my emails is full of junk spam emails. I never really unsubscribe from emails either because I've read a few times where you have to be subscribed to win, I can't remember the particular contests though.
But, I also use my search bar (sometimes, I guess I should more) and hunt for 'winner', 'win', 'won','congratulations'...( some keywords describing a win basically).
Now, I don't read emails because I only enter the major sweeps usually.. I'm planning to try & enter more in the future so I'll just use my search bar. Because ultimately if I read the subject line and rely on that. But, like you say it would be terrible to miss out on a win from not reading an email so it is up to you! That's my advice!
Oh yeah, let's say you have 2 hours of free time a day, well if you spend thirty minutes reading emails that decreases your entering time. After all, one cannot spend every waking moment sweeping usually people have chores, sleep especially, and eating, and appointments such as commitments, plus family time, tv time, kids, a job, etc. One entry can be the winner so If I were you I'd just balance my time accordingly. But I have read on different times where the email did not have something in the subject line for a win. I don't worry though, and I've never opened all emails or any really. Personally, I worry more about submitting my winner info, that's my big thing. I'm always in a rush so I wonder if I entered the info to ship stuff correctly lol...I do have an auto fill program though and it's 'last pass' and has never let me down as far as getting my prizes I've won.
Just don't forget the search bar!