I'm fine with the idea of a main panel with only 2 things in it feeding a sub-panel with everything in it.
Question #1 - I'd rethink putting a service panel outdoors at all. Weather is rough on panels, even if they claim to be outdoor rated. I'm a little nervous about a 100A breaker supplied from the normal bus bars, but if the manufacturer stands behind it, okay. The 100A wires are going to be a mother to wrestle onto that 100A breaker. Are you quite sure the power company has provisioned you 125A service? 100A is more common.
Question #1 (the second): You're gonna want more slots than 24, since this box powers pretty much your whole house. Nobody ever installed an addition and went "Gosh, that job was sure made harder and more expensive by having too many slots in the panel". It's a false economy, especially since bigger boxes are often bundled with more breakers. Your house may be ok now, but do a kitchen remodel and lookout!
Question #2 (the second): Don't bond your grounds to random plumbing that happens to be going by. It's not code, and someday you might have a plumbing problem and the plumber replaces a downstream chunk of it with PVC. Whoops. Also, they've been upgrading customers to PLASTIC water meters. Double whoops. Bond properly and to code.
Question #3 (the third): Bond ground and neutral only in the (singular) main panel. As such, you need 4 wires between main and subpanel.
Just for your edification, it's only a sub-panel if it's fed from a main panel. If it's fed directly from a transformer, it's a main panel.
I have a rule: Nobody does things for no reason. 600A is a crazy amount of service, and if the reason for it isn't obvious, figure it out before you change anything. Once you understand that...
Your principle of feeding one subpanel off another has merit. However you must bridge over all the wires together - two hots, neutral and ground. You never pull over a hot and have its partner wires return via a different route, For a variety of reasons. This means if one wire in a cable run is dead, the whole cable run is useless and must be taken out of service entirely.
But I wouldn't focus on the cross-connection yet. I'd try to save this setup. I'm betting it's a termination problem not yet found - it's unusual for wires to just fail.
The bigger problem is: This work was all installed at once, right? Whatever damaged one thing could well be damaging others. Did you find corrosion on the connections you serviced? They're all that way. Don't wait til they fail, because a neutral failing is especially bad news.
As a temporary workaround, like, to get you to the weekend when you can fix this, you might move your 120V loads to the breaker spaces that are working. If you need more breakers, either share breakers (when legal) or buy duplex breakers.
I find almost all such problems are with terminations, perhaps in a splice box you haven't found yet. Don't just tighten lugs, de-energize it and take it all apart and really give it a once-over and fastidious cleaning. If it's aluminum wire (AA-8000-series is legal and OK), apply the anti-oxidation goop. Only then, look at the cable itself.
Somebody who puts in 600A service doesn't cheap out on the installation. There's a good chance your cable run is in conduit, which is designed to make cable replacement easy. On the other hand, it surprises me that cable would fail in conduit, so I'd be concerned maybe something penetrated the conduit, damaging both conduit and wire. Any wire damage seems vanishingly unlikely under a foundation, far more likely in a more accessible place (perhaps a recently accessed place - had any diggers around lately?)
And by the way, it's possible to splice underground direct-burial cable.
If the cable is in conduit, easily replacing wire is the whole point of conduit. They spent extra money as insurance for this situation - use it! Validate that the conduit is physically intact (not collapsed). The techy way is to energize the bad wire (only) and use a detector to find the wire break (by looking for EMFs). That tells you how far down to send a borescope (a plumber will have that if an electrician doesn't).
A low-tech way to inspect conduit is to disconnect the wire bundle at both ends and pull it an inch and see if it gives more than the normal resistance. The "Alexander the Great"** method is to pull the entire bundle, inspect, maybe borescope, fish and pull it back in, and see what happens. Mind you, this is not to continue the old wire bundle in service - but merely to use it as a "test dummy" to test whether the conduit is pullable. If there are any defects in the wire, replace all of them unless it is extremely clear what went wrong.
If the conduit is intact and the wire is bad, this is routine: pull and replace the wires. That job is easier with specialized tools an electrician will have those on his truck.
Wires don't just randomly fail in conduit. This for sure: whatever killed the one phase threatens the others. You'll want to know what that is.
Best Answer
Start by shutting off the breaker in the main panel that feeds the subpanel.
Using a multimeter set to check continuity, test the continuity through the main breaker in the subpanel. It's possible that the main breaker in the panel is bad, and needs to be replaced. The problem is, it's usually cheaper to buy a whole new panel then to get a replacement main breaker. If this is a new panel, and the breaker tests bad. I'd contact the place you purchased it from, and ask them what your options are.
While the power is still off to the subpanel.
Check all the connections to make sure they are tight.
Check the continuity from the bus bar through each branch circuit breaker.
Make sure the main breaker in the subpanel is fully closed. If you turn it to the ON position and it has a little wiggle to the handle, this might mean the breaker is not actually setting when you turn it on. In the ON position, the handle should held tightly in place. If the breaker will not set, it means the breaker is bad and needs to be replaced.