[Nexus 7 (2013) Q&A] Johmama:
I am more than happy to answer that question for you. First: I want to shed a little light on what 'rooting' a device means, and what flashing new ROMs entails.

First and foremost: root is not required to flash new ROMs. Having root access to the file system is having full control and rights to your files. Think of it like having an Administrator account in Windows. You're free to manipulate all files on your device, including core system files, which is why you don't have root access by default; the normal end-user wouldn't know which system files do what and could delete a very important file that could soft-brick their device, or could download a malicious app that would do the same (think of it as a virus in Windows)(Protip: if you only download apps from the Play store, you're good).

Second: the only thing required to flash a new ROM is your bootloader to be unlocked, and maybe a custom recovery is required, I'm not too familiar with the stock recovery and whether or not you can flash .zips from there. Unlocking your bootloader enables you to flash unsigned images to your device, a custom ROM in this case. Here's where the confusion happens: when you flash a new ROM, it overwrites your old ROM, including the superuser status you had. Most custom ROMs have root enabled by default, which is why it seems like your old root access carried over. Think of flashing a new ROM as re-installing Windows on your computer. All of your old profiles are erased and you need to set up new ones. Having root access is like setting up your Administrator profile and using it.

Third: the benefits of rooting. I personally use root specifically for Titanium Backup, Cerberus, ES File Explorer, and franco.Kernel updater. If you don't know what those apps do, I'll explain. Titanium Backup backs up all of my apps so when I flash a new ROM, I can easily restore them, instead of having to manually re-install them from the Play store which takes time and sometimes, data. TB needs root because it's installing apps on your device automatically (after you tell it to). You can see why you would need root there. Cerberus is an anti-theft app, which needs root to remotely activate sensors like GPS, camera, take screenshots, show messages, get device info, etc. Again, you could see why root would be needed there: security. ES File Explorer (any root explorer app) needs root because, well, so you can see and manipulate files that you need superuser status to do those things. I use it for many miscellaneous reasons such as renaming my nandroid backups. franco.Kernel updater I use to tweak my kernel settings.

Now, I'm not trying to convince you to root. That is a decision that should be made by you. Having root comes with potentially negative consequences. You could mess up your ROM. The chances are very slim, and very reversible (worst-case scenario: flash new ROM), but still very real. If you don't know your way around the bootloader, or how to restore factory images to the device, it could be frustrating to fix a soft-brick caused by system files being manipulated. Still, you have XDA, full of helpful people and plenty of guides and if you have the patience to read and listen to advice, you should be able to learn how to fix your problem, should one arise.

Ok, sum-up. Having "root" is like having Administrator rights on Windows. Installing ROMs is like re-installing Windows on your computer. Recovery is like the BIOS on your computer. Therefor, root is not required to flash a new ROM, only your bootloader unlocked (possibly custom recovery). Root is nice to have, and poses almost zero risk to hard-bricking your device. Having root could soft-brick your device if you don't know what you're doing, however, and could potentially be very frustrating if you don't know your way around the bootloader and/or recovery. If you have the patience to learn or listen to advice, it can be fixed though. It's up to you to weigh the risks and rewards. One thing to keep in mind: if you only download apps from the Play store, you should be 100% A-Ok to have root. The only way to soft-brick your device then, is if you're in there poking around with files you shouldn't be messing with (so, user-error). If you do decide to root and/or flash new ROM's, I'd advise in doing it manually at first instead of using toolkits. That way, you can see what's happening, which could come in handy if something should come up later.