I personally don't think the attack titan has the power to see into the future. I believe this association only comes because the future holder of the attack titan is Eren, and unless we hear actual evidence from a previous user from the great titan wars, we'll never know, but surely its secret would have been made known or used better previously?
Don't forget that memories can be manipulated by anyone with the founding titan and they don't need to be titan shifters for it to occur. I think that Eren simply manipulated Grisha because Grisha was the one who held the key to Eren's power, and the people before Grisha all had the Attack Titan in common, so it makes sense that they'd also get memories if Eren wishes (Owl knowing about Armin). We also know that in S1E1 Eren woke up from a nightmare, presumably the start of his perception of seeing time all at once; he sees his mother dying, then it actually happens in front of his eyes. At that point in time we thought it was tragic that he couldn't save his mother, but knowing that he may have had a nightmare fortelling that specific event would compound Eren's grief and stoke his rage against the titan's. If it was the attack titan's power for Eren to see the future, why would Eren have received that nightmare / message from the Founder Ymir + Future Eren before getting his Titan?
So what is the Attack Titan's "Power"? Well, I honestly think its aggression, plain and simple, is enough. The armoured titan does have armour, but the attack titan proved better in combat and beat him. He also beat Annie and the hammer titan too - it doesn't need a special power if its tenacity and aggression is enough to fulfill its role as the vanguard.
Final speculation: Eren's "TATAKAE" mentality suits the attack titan perfectly.. I wonder why? We know that Ymir wanted to be freed from enslavement and worked with Eren's future towards her freedom, and yet the Attack Titan has always been about freedom and fighting, according to Owl. I wonder if this situation has ever happened in the past, or if Eren's influence spreads further into the past. Or maybe its just a coincidence that Ymir's wish, Eren's nature, and the Attack Titan's legend all mix together?
Thanks for reading ❤️
Note I wrote this on a YouTube comment and thought I'd post it here since it's probably not gonna be seen by anyone lmao.