This reminds me of my situation and rival, we even have the same colors! Except my rival's falco is too good. >.> That falco wasn't that good. Lol @ edgeguarding with phantasm.
Things to improve:
1. Sweetspot the edge. Doing so will minimize risk of getting knocked out when you could be hanging onto the ledge, surviving. I still haven't got it down but that falco definetly took advantage of your non sweetspotted up b's.
2. Use the up tilt to continue combos, not always to start them. I saw you use bair out of a lot of comboes, which generally hit them too far away to continue them. Not necessarily always use uptilt instead but thing of other things to use besides that bair. Its not that good unless you're edgeguarding/near end of stage.
3. Gay falco. Maybe you haven't played against a lot of falco/fox, but their recovery is very predictable. Especially if they're recovering under the stage. Space yourself, and use dtilt/fsmash, or counter sometimes if you can't get the sweetspot or space yourself fast enough. Dair also if you're ok at it. Also, you can backthrow them off, but if you're relying on a shield grab make sure to keep an eye out for the forward smash.