Restoring from a TM disc doesn't require more space... only enough space on the internal drive to restore.
I don't understand why you thought that was a "good idea" in the first paragraph if the Mac to which you restored was the same Mac from which you were backing up to the original TM drive. Conceptually, the Mac already had all the files on it that it should and that's what you were then backing up to the original TM drive. A restore from the TM drive wouldn't do anything since- in a restore process- it is only looking to put files back that are not there. All files you wanted to back up should have been there, so no restore should be necessary. I would guess that a TM restore would end up doing nothing in such a scenario, since it wouldn't find files missing to restore.
The right thing to have done was to simply plug in the new 4TB, select it as your new TM drive and let it back up that same Mac. It would essentially be taking over as your TM backup disc.
Your description is confusing. One way I read it is like you've got a new Mac (perhaps different than the Mac mini you referenced in post #1). Post #1 reads like you have a 2023 Mac mini backed up to a TM disc and the TM disc is showing that it is full. Post #15 seems to imply there is a "new computer" with consideration to "do a backup directly from the old [computer?]."
If there is a new computer, instead of a restore from another computer's TM backup, you should be using a different program called "migration assistant" (in the utilities folder) to then import from the TM backup to the new computer, not TM. If you have restored one Mac to another instead of using migration assistant, that fits your description of "looks different", "apps & files are missing", etc... and possibly explains a "not enough room" message too. And if this, the remedy is likely:
- Reset the new Mac to new again (basically wiping anything restored to it)
- Boot into it like a brand new Mac owner would.
- There will be a screen in the setup process that will ask if you want to migrate from another computer or a Time Machine backup. You can choose either and follow the instructions. That will NOT be restoring a TM backup from another computer but running "Migration Assistant". What you want to happen is the same but how it does it is different.
- That process may take a while so just let it do its thing.
- When it is done and you finish the steps associated with firing up a new Mac, it may still look different if the new Mac is running a newer macOS than the old Mac. But your files & apps should generally be where you expect them to be.
If there is no new computer and all this is about the 2023 Mini, while you should NOT have restored to it if it was as you wanted it before the restore (no benefit to that at all), if you have and now it looks different, you'll need to share more information. Assuming the TM backup was from this
same computer, a restore to this computer should not alter apps & files. In short: it should look "the same" as it did when you did the last backup to that drive. It may require another restore but the outcome you describe doesn't fit the logic of how TM "restore" works.
If there are 2 Macs involved plus one TM backup drive, I suggest throwing all of that in the car and heading for an Apple Store for in-store help. You may be getting into complications that are going beyond what strangers are able to help you do relative to your own knowledge & experience with all of this. In store, they can reset the new Mac "to factory" and then help you run the initial boot and "migrate" your information from either the Mac Mini or the TM backup.