Well, playing aggressive isn't a bad thing whatsoever. I do agree that many Falcons have the problem of using unsafe moves during pressure over and over. My personal strategy is to pressure with moves like SHFFL Nair / Bair on the ground; moves that are generally safe and quickly canceled upon landing. They're good at racking up damage at low percentages because they combo into other moves (i.e SHFFL Nair > Up+B/Jab/Grab and Bair can be used to get a buffered dash grab) and if it gets shielded, you're still out of range if you were poking with the moves.
Of course, using the same moves over and over isn't a strategy. In fact, it's stupid and people will eventually begin capitalizing on punishing those specific options.
That's where the grab game comes in. Depending on the spacing between you an the opponent, Grab, DashGrab and PivotGrab can get around defensive playstyles when the blatant pressure isn't working. The grab game is also our most reliable combo starter beyond 50% in my opinion (i.e. Dthrow > Uair > Knee is a true combo). Using it to keep in a character's comfort zone and punish reactions is essential; especially due to the properties of Falcon's dash grab in this game. The distance boost that Falcon gains from his dash grab animation is almost on the tier of a tether grab without the long and punishable cooldown. You can grab characters that are a short distance away in a short amount of frames. Only problem is that timing it incorrectly can result in you sliding past the opponent. Knowing when to use a dash grab vs. a dashing pivot grab makes all the difference when playing Falcon.
When someone is in the air, it's needless to say that our Uair is essentially our best move. Again, I capitalize on this factor. Rather than jumping up and Uairing people senselessly every opportunity I get, I short hop under opponents when they've used their second jump and are close to landing. This puts many characters in awkward positions where they have to choose to try and trade hits with Uair or try to air dodge. And if they air dodge, you get a free hit in. SHFFL Uair will allow you to recover just as quickly as they do from their air dodge, except you're landing first, so you can punish their air dodge accordingly. This is where I feel that Falcon mains fall short.
It's also worth noting that Falcon's Jab is an extremely useful move for shutting down many approaches if you're taking the defensive/passive playstyle. For example, a Sonic begins spinning toward you with down+B or side+B; simply hold jab in place and he'll get hit if he comes in contact with you. When I'm playing against Sonics, they've always had a habit of trying to spindash fakeout > Homing Attack when I've used jab. But of course, Falcon can simply stop jabbing and Up+B the Homing Attack. If a non-beam projectile is coming toward you, jab will generally cancel it out. There are a few projectiles that this doesn't apply to, but knowing the matchup and what to use it against can make all the difference.