OK, need to make some things clear, and then provide you with a process to get where you want to be. This will be somewhat long, but the detailed explanation is necessary.
1. First, can you boot the iMac up? And if you can, what can you do with it?
2. Secondly, assuming you can boot up the iMac, once it gets to the desktop, go to the Black Apple Icon/Symbol in the upper left hand corner, click on it, and select "About This Mac". The next screen you'll see will contain the Mac OS information (plus some other details).
3. I suspect you have done little, if any, disk cleanup/maintenance/repairs, from a software perspective. Is that accurate? Also, you can do a good amount of disk cleanup on your own, and there are some excellent products available (both free and commercial) that can help you with those tasks.
4. Additionally, given that you have done OS "upgrades to newer" ones "in place", the stuff on your internal drive (no matter which account) most likely is not "lean and clean".
5. Just to get the terminology correct, you are not booting your Mac from the backup, with what you described above. You are booting your Mac from the (hidden) Recovery HD partition on your iMac (that's a small partition created by an installation (not upgrade) of the Mac OS). From that partition, you can do the following tasks:
1. Use Disk Utility to 1) check out/repair the internal drive on the iMac, and 2) do an Erase and Format of that drive.
2. Do a clean, fresh, "virgin" installation of the "applicable" Mac OS (I'll explain that below).
3. After that installation, you can then get to the Time Machine backup, and "migrate"/copy needed "stuff" from the backup to your iMac's drive.
In fact, those are the steps you need to follow (although it might be best to do one more Time Machine backup, just to make sure you have all your latest information).
Now, a couple of "caveats" when you do that:
A. Hopefully, Disk Utility will find no issues with the drive. If Disk Utility finds any serious issues (the software will tell you), you cannot proceed any further. Also, when you use Disk Utility to Erase and Format the drive, make sure you select the "top" (ie, Volume) level of the internal drive.
B. I am unsure as to what exact Mac OS would be installed via step 2 above. As it is, you will of course need an internet connection, and the "applicable" Mac OS will be selected from Apple's servers, and the installation will occur over the internet. That will not be a fast process (and of course dependent upon the speed of your internet connection). If it is Yosemite (V10.10.5), then you should be OK. But, if it is a later one, then you'll need to insure that whatever third party (ie, non-Apple) apps you use are compatible with that newer OS. That could entail needing to download needed updates.
C. Finally, when the OS installation completes, you will be offered, via Migration Assistant, the opportunity to "migrate"/copy needed "stuff" from your backup. I am just about certain that you can select your account only to do that copying, but I'm not 100% sure, as I only have one account on each of my Macs. (Note that this step will 1) copy everything from your account, including any unnecessary/redundant stuff (due to not doing disk cleanup/maintenance/repairs), and 2) result in NONE of your son's stuff on the drive.).
If you want to know a faster, "cleaner" and easier way to do all that, just let me know, and I can provide details as to how I use SuperDuper! (and some other things) to do the same process, in a much faster (and more controlled) fashion.
Good luck!
