It wouldn't be absolute. It's possible, likely even that the subs won't be picked up by the ordinary radars, but they'll get the occasional spot that way. Same with air-to-surface radar, a sub won't be picked up nearly as easily, particularly in bad weather, but they'll get some spots. The sub will have to surface, at least partially, to line up attacks, allowing for the eyeball to spot a few submarines, and some torpedos to be dodged.
Basic depth charges are also very easy to make, you can modify an airplane general purpose bomb into one easily, and then just kick it over the side. It'll be almost completely ineffective, but technically that's a depth charge.
In general, I'd expect them to pick up a few kills with air-to-surface missiles, when the luck factors line up, but the fact they have radar on the missiles themselves makes it much easier for the sub to differentiate between there being enemy radars in the area, and when someone's actually shooting at them, and get underwater safely. Particularly because of the long range of missiles in general in this theater.. if the target isn't there by the time the missile arrives, it won't get hit.
Air ASW will be heavily hampered by the all-missile loadouts, as a conventional bomb or even torpedo can act as a depth charge much easier than an air to sea missile, (even if this weren't the case, an ad-hoc one would be only slightly more useful than one on a ship), and the great speed and low loiter time for jet aircraft will make it basically impossible for planes to hunt after submerged subs.. even 15 or 20 minutes underwater would mean the plane would have to start over from square one. The restricted aspects caused by the high speed and missile combat would also help things, as I'd expect the plane to not get a proper aspect and have to take another pass for a second shot, by which time the sub would have plenty of time to submerge.
So yeah, I'd expect them to get a few kills, from the occasional missile hit and a few from gunfire, but they would indeed be more or less incapable of sinking a submerged sub, and wouldn't really be able to take out too many even in transit on the surface. If they stripped the guns from their smaller ships, which they might do or have done, I'd expect things to be really brutal.. a submarine that sneaks into point-blank range in a formation could sink a gun-less missile destroyer with torpedos, without taking a scratch. I'd also have them hunt for merchants, with ASMs the big threat I doubt the merchants are armed with deck guns, rendering them only so much meat for the subs, and possibly reducing their resource flow for a turn. No better way to keep a push going than to raise the expense rate of some enemy designs.