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Moving the filesystem root to a different partition and booting from it

My primary drive is partitioned basically like so (only roughly to scale):
```
 / /home                                       junk
 v v                                           v
|xx############################################*************xxx
^                                                             ^
/boot/efi                                                  swap
```
<details><summary>Motivation</summary>

The "junk" space is in fact separately partitioned and formatted - it's left over from a failed attempt to set up dual boot with Windows, but I no longer have any interest in that. I don't care about anything written on these partitions and would be happy to wipe and reformat them and reuse the space.

The problem is that the partition where `/` is mounted is far too small for my needs. When I initially set up Linux, I followed old advice with unrealistic expectations about the "system" partition size (and allocated a larger swap partition than I really need). Essentially: I want to be able to install more apps, run the distro's OS update rather than reinstalling for new versions, and not worry so much about maintenance tasks like `apt` cache cleanup or `journalctl` vacuuming.

So I want the OS to be on a larger partition, and fortunately I have tons of room to work with. But I don't want to move or otherwise disrupt the `/home` partition for this, first off because it's hundreds of GB and second because I don't want to risk needing to reach for a backup for all my commonly used data.

I've come up with a plan where I won't have to touch the `/home` partition at all. I might expand it into the "junk" space in the future, but I definitely don't want to move it. I might also try using that space to install separate Linux distros side by side - haven't decided yet.
</details>

My current plan is to:

* Back up everything anyway
* From a live boot, use Gparted to, in order:
    * scrub the junk partitions
    * expand the current swap partition to the size I want, and format it as `ext4`
    * copy and paste the `/` partition to the former swap partition
* Set up the system so that it (will|can) boot from the new `/`
* Reboot from the new partition
* Use Gparted again to wipe the old `/` partition and designate it as swap space, and start using it

**My primary question is about the boot setup step.** How can I reconfigure the system so that it will boot assuming that `/` is mounted on the new partition, and ignoring the contents of the old partition?

Aside from that, I want to make sure that I'm managing the swap space in a valid way. I don't want to erase the old partition until I've confirmed I can boot from the new one, of course. Will Linux complain if I try to boot the system without either a swap partition or a swap file, even if there's plenty of RAM for basic tasks? (I.e. will I need to create a temporary swap file?) I know I can `swapon` the new swap partition for that boot once everything else is working, but how do I make it permanent? Finally, with my current plan, can I avoid the need to start with a live boot, if I `swapoff` first?

If it matters, the OS is Linux Mint 20.3 (Ubuntu 20.04) running Cinnamon.