What’s a practical way to centralize, access, and back up family photos from multiple phones and a DSLR?

I think there are a lot of solutions, but I can't say I know the best one. As you use Apple devices mostly I'll describe the full Apple solution below.

You indicate that this seems like a bad idea. But given that you use Apple devices mainly I think it's the path of least resistance (neglecting cost).

Please note that I don't have mobile Apple devices, nor do I have a (shared or paid) iCloud account, so I might miss some steps or limitations. I've just pulled it together from internet resources. In the comments it became clear that it might not be possible (anymore) to have multiple iCloud accounts. This recent-ish article indicates you can, but I unfortunately cannot test it.

Steps to go full Apple

1. Get another paid iCloud account. This will serve as a photo-only account shared with your wife. You now have 3 iCloud accounts:
   • iCloud James
   • iCloud wife
   • iCloud photos
2. Turn off all sync options except photos for the iCloud photos account on all iPhones and iPads. It seems you have to set the iCloud photos as your main account.
3. Turn on all sync options except photos on the iCloud James and iCloud wife accounts on the respective iPhones. For this you have to add iCloud James and iCloud wife as a "regular" account (like Google, Outlook, etc...).
   • This allows you to keep your mail/contacts/notes/etc... separate from your wife.

Now all i-devices can sync and view each others photos. This answer on Apple.SE describes more or less above solution.

Only portable device left is the DSLR. I'm afraid you'll have to manually transfer those photos from the memory card into the iCloud photos account.

4. Setup the iCloud photos account on the laptop.
5. After taking photos with the DSLR copy all photos to the iCloud Photos folder on the laptop
6. After copied successfully, delete all photos from the memory card. This ensures that you will no longer wonder if files on the card are already synced or not. Files on the card means that they're not synced.
   • Last two steps could be automated with a script, but that's outside of the scope of this answer.
   • This could probably be automated with a raspberry pi and a card reader, circumventing the slow laptop

Pro's

• Everytime your wife or you take a picture with your iPhone, it's synced with the iCloud photos account, making that account the single source of truth.
• All photos are easily accessible via the iOS interface.
• You probably don't need 3 paid iCloud accounts as you now (presumably) don't need much space on the James and wife iCloud accounts, allowing you to cancel or scale down those accounts.
• You get rid of the generic cloud storage.

Con's

• It's all Apple. They can raise prices, cancel the service or change it any way they see fit.
• The DSLR part remains either manual or requires some robust scripts to automagically push memory card contents to iCloud.
• There's no back-up. If any device on the iCloud photos account deletes all photos by mistake, this mistake will be synced across all devices leading to the loss of all photos.
  • This is inherent to syncing services and can be solved by introducing a real back-up service that takes periodic snapshots of your iCloud photos account. This can e.g. be a combination of your old laptop, some back-up software and a large external drive. See also this previous question regarding backup