Even a few years ago it felt like you could make the case that Steve's physical feats (strength, endurance, durability, reactions etc) were a cut above Bruce's.
I'm not sure that's the case anymore. If anything the opposite may be true.
In my opinion, Batman on average can rise to Captain America's level, but Captain America can rise to a level above Batman's average if that makes sense. Writers that work on Captain America can always one up Batman because Batman on average has always been written as peak human, while Captain America is a Super Soldier, which gives them more leeway to score a bit higher in terms of pulling off power stunts.
Batman can rise in terms of physical feats to Captain America level, and he has done things that are clearly super human (kicking trees, yanking out metal doors, getting bodied by Diana without his face turning to mush).
However Captain America can definitely amp things to the next level beyond what Bruce can do. Sure, Bruce can blitz nearby enemies, but Steve can run across the battlefield carrying someone at high speed. Sure, Bruce can fight enemies for days on end with no rest until his body finally gives up just in time for Bane to break his back. But Steve will never tire.
So, physically Bruce can keep up with Steve, and for all intents and purposes they can seem ‘equal’ to an outside observer. Accuracy with thrown projectiles may look similar. Fighting capability may look comparable.
But Steve can dial things up and (for example) throw a shield at a missile that is already flying and the shield catches up and brings it down, or throwing a shield through a truck ....
Batman is scary strong, but Steve is superhuman.
WHERE BATMAN comes in ...
... his mind and resources. He’s a medley of a Captain America, Black Panther and Tony Stark, and that is what enables him to punch above his weight.
Eg. a suit design (where he was helped by his JLA team mates) that can face off against Darkseid, another suit he made all by himself to defeat the JLA that had miniature red suns in the gauntlets (no, not red solar energy blasts like Lex Luthor would have ...actual miniaturized red suns in the knuckle sections), and obviously his prep ‘powers.’
So, mentally that’s where Batman brings the bacon - his mind. Otherwise there are other characters (eg Shiva in skill, Deathstroke in abilities, David Cain in tactical preparation, Mr Terrific in intellect, Lex in strategic resources, Tim Drake in war tech, Constantine in knowledge of the mystic - Zatanna has theorized Bruce could be Homo Magi - etc etc). The only way the Mary Sue called Batman can be in the JLA - where every character is a god of sort - is his mind.
So that’s his key advantage.
But in terms of physical abilities - Steve is his superior.
He has done that and more - eg the sneaking up on people have been on the Spectre as you mentioned, as well as White Martians, Kryptonians, the likes of Gotham and Gotham Girl, the likes of Wraith in Superman Unchained, etc.
But I see those as part of his ‘skills and abilities’ rather than raw unabated physical ability.
Eg, in a lifting competition Steve could outdo Bruce. Same in a running competition, or anything that would be in the Olympics.
It’s just that Bruce brings much more than just physicality as we both mentioned. Much much more, at almost ridiculous levels.