Not sure of all the details you're facing, so I'll give a more generic response.
Today, we install services where there is a disconnect with a fuse / breaker right at the meter. This point is the "service," and is where the neutral and ground are to be bonded.
Older services are often a simple meter pan, with no way to disconnect power without pulling the meter. Others can argue whether this is 'legal' or not today- as far as I'm concerned, this is such a bad design that I won't do anything until it's changed.
Assuming that you have a disconnect at the meter, as well as overcurrent protection there, everything downstream is a "sub-panel." Keeping in mind the tap rules and feeder ampacities, there are many ways to power multiple panels. You can start with a panel at the service; you can feed through each and every panel until you get to the last one; or you can mix the two approaches.
The devil is also in the details. Very few load centers come with lugs that have any provision for multiple conductors; landing two wires under a lug just because they fit is not allowed.
Then there are those issues regarding bonding of the neutrals. Since few older panels have ground busses, and the neutral bus is often bonded to the case, you have an arrangement that cannot be made compliant with today's rules.
Under today's rules, you can have all manner of panel arrangements. For example, my own house was rewired with the following layout:
Meter feeds a main panel. Grounding electrode conductors and the neutral bond are in this panel.
The main panel feeds a local receptacle, as well as three other panels: the inside "primary" panel, the outside HVAC panel, and the outside deck panel.
The inside primary panel, in turn, has branch circuits- as well as feeding the kitchen panel and the bath/laundry panel.
The HVAC panel serves the mini-split air conditioner units (there are three), as well as a tiny panel in the carport storage area.
The deck panel serves the deck, as well as the (planned) hot tub, storage shed, and workshop.