The Nats actually had a discount last year for early birds during the playoffs. For the first hour after gates opened food and beverages were 20% off.
My thought was that that discount wasn't enough to get a significant number of fans there early. Basically meant that Nats Park was less over priced than normal. People looking to save money aren't going for that discount. An NLDS branded giveaway or a potential to win autographed items would be a much bigger draw.
Early bird/pregame attendance was fairly good on the Saturday (4:30 start or thereabout) but it's always going to be hard to get people down there much before 5 or 5:30 pm on a weekday.
To entice people to leave work early, you need to give them a decent shot at getting whatever the giveaway item might be, which means having a lot of that item. If there's only a few thousand, then people won't bother, figuring the item will all be gone before they can get there. And if you need 25k NLDS hats or whatever, you need to get that order in a few months in advance, which no team will do because they're trash if you don't make the post-season. MLB isn't going to furnish them because why would they. If we have another 5:30pm weeknight start (counting Friday as a weeknight since it's a work day for most folks), they need to just focus on crowd/traffic management, with the peak entry being between 4:30 and 5:30. People will leave work a little early to get there and settle in but they won't scoot out at 2:00.
