Without seeing your swing, I can't really say what might be causing your push-fade. However, in relation to your question, I have seen players hit push shots without an in-to-out swing path. That being said, the position of the clubface at impact has an undeniable impact on both the line your ball starts on and the path is travels thereafter. One important thing to remember is that your body, assuming you have any athleticism, will subconsciously alter itself to try and make up for a clubface that is too open coming into impact--that is, your brain picks up that if your clubface strikes the ball in the open position it is traveling on, the ball is not going to take the desired route. So, in a last-minute effort to fix it, your brain tells your body to quickly change something to put that ball in flight. Moreover, this last-minute change can be a little different every time, resulting in inconsistent ball striking....which includes your push-fade. This quick, unpredictable swing alteration is obviously undesirable. So, you need to focus to putting your body, the shaft, and the clubface in the proper positions approaching impact to avoid your subconscious reactions taking over.
Without seeing your swing, I will take a shot in the dark and recommend three things to try:
(1) Between the top of the backswing and impact, there is point every golfer shifts from "letting the club drop" into "I need to start releasing this club so I can get some power behind this shot." Focus on that shift happening a lot later than it is happening now. Really emphasize a delay. It is hard, but it works....and it cures numerous swing faults; e.g., casting, insufficient weight shift, loss of spine angle, lack of balance, etc. Isn't everything easier when you have more time?
(2) Focus on hitting down on the ball. This cures many swing faults as well, and it's a lot harder to hit a push-fade when you are taking a divot AFTER impact and compressing the golf ball. I would recommend doing this not only with your irons, but your fairway woods and driver as well; you might not really be hitting down with your woods, but the swing thought will help acheive more consistent strikes.
(3) Finally, make sure your right leg is angled in more at address than your left leg. To do this, feel like you pop your hips to the left at little bit at setup, and lower your right shoulder a tad to help counter-balance. After that, put all of your focus on keeping your right knee angled in during the backswing; don't let it sway to the right, and try not to let it rotate (it should actually rotate some, but you don't have to force it because it will inevitably happen whether you like it or not).
PS- Don't try all of these at the same time; instead, try them separately and see if any of them help. Plus, that's way too much to think about at once.
Let me know if any of these help.