I think with really good SDI on FD the Fox can milk some kind of slim advantage in terms of heavy punishment because his chain grab can't be SDIed, but the margin for error with Fox for reaping a strong play off a shine is way lower since tech chasing off a knockdown has a lot of factors that work together and he's far more likely to get ***** if he flubs and gets shined himself or such. His other combo starters have the weakness of being weak (dash attack, nair) or prone to inconsistencies because of DI abuse (edgecanceling an u-smash, SDI dair). As a side note, notice how bair never comes up? If only he could bair forwards :x
That said, I think on FD I'd still probably say a controlling Falco probably has the advantage despite Fox's chain grab (and Falco can easily still hit really good combos by modifying his combo tree very slightly vs SDI on certain things and a lot of it has very lenient reaction windows). The advantage he gets in neutral by the lack of platforms for Fox to maneuver around lasers gives him a lot of ways to force Fox into full jumps or to limit his ground movement significantly (PS helps here, but Fox's low range on grab can lead to dumb stuff happening if Falco buffer sidesteps on the grab or shines immediately after being lasered, etc). Fox's so-so priority in front of him is a real problem here because Falco can influence which direction he faces during low movement, and his priority below is soft to Falco's quality bair, u-tilt, and so forth. On that note, jumping into shine with Falco can be super amazing and Falco's shine in pressure on shield is also generally stronger because he can combo off stray shine hits, whereas Fox can't reliably.
I don't think Fox is really that disadvantaged overall, I just think Falco on FD is special and compromises a lot of Fox's best tools in the matchup (from the platforms he can zone with the threat of bair and pretend to be a mini-Peach with a crazy repositioning game, a good DJ, the shine, and an amazing u-smash). To the Europeans in the audience, I apologize if I sound dumb because I admittedly don't know much about Fox's PS and everyone in Europe seems to be able to PS with at least 85% consistency rate. So if I missed the boat and he can actually just pretend to be Marth and dash PS grab into **** as a gimmick so strong it's in actuality a mediocre strategy that adds convenient cohesion to other parts of an anti-laser repertoire then cool.
As a side note, a funny thing I've noticed about Falco is that I always find I consider Falco's hardest (read: even) matchups as the ones where the bird often dies in one hit and doesn't hit the opponent as hard relatively speaking (Sheik, Marth, Peach). Falco's opponent also needs enough range to outplay him in the neutral effectively once his pistol has stopped working. Fox is less good at death comboing Falco at high level than those three and he takes stupid damage from the moves Falco fishes for (shine, u-tilt, dair on grounded opponent, nair on airborne one).