Was this installed to code and is it still to code?
Yeah, it’s a mess alright.  The combining of neutral and ground in the main panel isn’t wrong, though.

First, the problem of the 60A wire completely unprotected, since it’s on the meter side of the main breaker.  If that cable took a nail or something, it could flow as much as 10KA and nothing would trip.  Nothing.  The service drop would glow cherry red, but this 6/4 cordage would glow cherry red first and set the building on fire.

Buildings take a calculated risk with the service entrance from the weatherhead to the meter to the main breaker. But by design, this is kept to a bare minimum.  Your electrician extended this “logic” to that disconnect switch, which would be a pointless thing except for the above problem.

The double-taps of the breaker.  Regardless of the above, sticking 2 wires on a lug only listed for one is always a huge no-no.  Now, they make double lugs, and they also make 3-void Polaris connectors.  A reason to do that legally would be if you decided to add a second 60A subpanel fed off a 60A subpanel.  No need to fit a 60A breaker in the first sub; simply tee off the first subpanel’s feeder.

cordage being used as a substitute for building wiring.  The giveaway that it’s cordage is “/4” which is cordage-talk for black white red green.  The ground wire is counted as a conductor.  In house wiring, it’s called /3, because the ground wire is taken for granted.

The disconnect switch lacks appropriate working space, obviously... but who cares? It’s leaving.


So how do we attack this?
It’s a Gould panel, but it’s also I-T-E, and their direct-lineage descent is Siemens. (Who also got the Murray line).  So Siemens kinda has 2 separate lines of breakers, and it looks like their Q or QP line ought to do it. Hopefully our panel expert ThreePhaseEel will come by and give us a definite answer.
However, I’m unclear on whether this panel is rated for tandem/duplex/double-stuff breakers. I can see that it’s a CTL panel, meaning the post-1966 CTL limitations on double-stuffs do apply to it.  Generally when they accept them, they say so, or at least hint at it, with a model number like Foo2032bar (meaning 20 spaces 32 circuits).   We get nothing here, so let’s assume the worst unless ThreePhaseEel can tell us otherwise.
The first rule of grandfathering is you can’t make a situation worse.  So we identify every circuit that serves something which must be a dedicated circuit today.  That’s laundry room receps, bathroom receps, kitchen receps, dishwasher, disposal, built-in microwave.  Put red stickers on those breakers; can’t use ‘em.
Also look for any red wires in cables; if you see that, identify the cable and follow BOTH the Red AND Black wires to their breakers and red-sticker both. Those are MWBCs, we don’t wanna mess with them, except it’d be good to put them on a 2-pole breaker.
Now, we identify two circuits with the same breaker trip value (15 or 20), that could stand to share a breaker because they aren’t used very often together, or they don’t have much load. Pull both their leads off and pigtail them to 1 of the breakers.  Do this twice and we have 2 free spaces.
Fit a 60A 2-pole breaker.  My wild-guess is a Siemens Q260 and hopefully ThreePhaseEel can guide us further, but most likely an Eaton CL250 would be the universal substitute (but only 50A).  Don’t use Eaton BR, Square D or GE.
Now run a 6/3 NM-B (Romex) cable from this new breaker to the main lugs of the subpanel.  Bypass the disconnect switch.
I think that licks all the problems!