OK here goes... (please take a deep breath)
I'm going to include TONS of details, hoping that something here might help you get to the answer.
If it's too much for your brain at whatever moment you get time to read all this, I do apologize.
Maybe come back later when they're sleeping and you're wide awake, if that is ever possible. I was a stay-at-home Dad for one precocious child for a few years, so I have a tiny idea what you're talking about, though our married friends with multiple (usually bickering) children always assured us that having just one child is nothing at all like having children. So bless you! My heart goes out to you!
Anyway, in Windows, there used to be a generic HPPCL driver and a generic PostScript driver for older printers. You could usually use a "generic" driver just fine, because very few programs (other than page layout and graphic design) used even 1/10th of the capabilities of the lowest-common-denominator oldest version PCL or PostScript command set. Something like that DOES exist in the Mavericks print installation dialogue.
But back to YOUR printer.
I tried installing my HP LaserJet 1320 on my iMac running Mavericks*.
Unlike Windows, you can't install a printer that OS X Mavericks cannot 'see', either directly connected or via a network 'share', or wirelessly.
My HP is connected via a network share. This should not significantly change what happens when you install your printer, but no guarantees.
This was my first attempt to install the HP, because my default printer is a Canon inkjet, connected wirelessly via WiFi.
So the LaserJet is in another room physically connected via USB to a MacBook, and marked 'Share this printer' on the MacBook system preferences. The MacBook has to be awake (not sleeping) and in this case, physically connected for my iMac to 'see' it.
So, when I clicked the + sign in the , System Preferences, Printers & Scanners to add a new printer,
I entered the description HP LaserJet 1320 into the search box the first time, then
1. Mavericks tried to "find" the HP LaserJet on the network, connected to the MacBook.
2. "This printer requires software that is not installed..."
3. "Downloading Printer Software Update..."
4. "Setting up "HP LaserJet 1320 Series @ KoMtB (MacBook's network name)
5. At that point the printer appeared in this Add window shown below:
Note the bottom Use: window below Location...
Note that the third choice is Generic PostScript Printer...
THIS IS THE ONLY PLACE AND THE ONLY TIME you can apparently select anything about which printer 'language' the Mac is going to use to talk to your printer:
In searching for about 10 minutes, I did not find any way to ever get BACK to this screen once the printer installation is complete, short of deleting and reinstalling the printer again, or adding the printer a second time with a different name.
I selected the choice highlighted, HP LaserJet 1320 series. I did a test print from TextEdit, and it worked perfectly the first time. I also tried it from Pages, and again, it worked perfectly, no spewing of PostScript initialization strings...
My suspicion is that maybe Generic PostScript Printer is getting selected by Mavericks somehow without ever informing the user.
Maybe 'Auto Select' does it. Don't know.
In any case, since you've already done a Printing System reset, you have nothing else (custom print job Presets) to lose by trying to reinstall it again.
I tried it, and It SHOULD also be possible to install the same printer multiple times, PROVIDED THAT you rename each one slightly during the Add process (HP-3200, HP-Auto Select, HP-Generic PostScript... ), so you can give several of them a try until you find one that actually works.
Now, here's how to find the 'generic' PCL printer driver...
Add the printer again, giving is a new name.
In Use: select "Select Software", then search for "HP LaserJet Series PCL 4/5"
HP LaserJet Series PCL 4/5 driver IS a 'generic' PCL driver, and SHOULD work for basic printing, provided that you DO NOT use the slash between 4/5 in the printer name use a dash - instead. When I tried it with a slash (probably because it was a network share, the printer did NOT install. When I tried again and used a dash, it worked fine.
The PCL 4/5 driver comes with a disclaimer as shown below:
Your HP is apparently from the year 2000? (Mine is from 2004.) It's so hard to give up old workhorse friends, especially when one toner costs almost a much as a new printer. The good news it that newer LaserJets are WAAAAAAY faster, and a toner for a new model still costs almost as much as a new printer... 😢
After years of buying ONLY genuine HP toner, I finally purchased a generic toner cartridge online last year when I needed to print about 1500 pages. The high capacity generic refill cartridge cost 1/3 of price of a genuine HP 'standard capacity', and it works beautifully.
Of course this is time consuming, especially when you have active kiddos competing for your time, but the last alternative after Gutenprint is to purchase a new printer.
*DISCLAIMERS & FOOTNOTES:
I don't use OS X Mavericks much, so I'm really not the best person to ask. But I did answer this thread ages ago.
So, if none of this helps, please post your question again as a brand new message in the OS X Mavericks forum.
That way it will likely get far more 'views' than posting your question in an old thread already marked 'solved'.
From what I've seen in the Apple Support Communities, those most knowledgable tireless Saints of Mac with tens of thousands of reputation points who frequent these forums most every day usually look for the 'unanswered' and 'unsolved' questions first.
Mostly I use OS X 10.8 Mountain Lion, because that's what shipped already installed on my iMac that I purchased from Apple as a refurb in Dec 2013. And because 10.8 seemed more 'sorted out' than Mavericks did last December.
When I got my iMac last December, I immediately installed OS X 10.9 Mavericks on an external drive that I could select at boot time because:
1) I could get Mavericks for free (wasn't sure if it would be a limited-time thing)
2) Purchase of a new or refurb Mac then included a free iWorks suite (Numbers, Pages, Keynote) latest versions (wasn't sure if it would be a limited-time thing)
3) I have Windows 7 installed under Parallels 9 on Mavericks to run basically one legacy Windows application I spent years learning and still use once or twice a year. Honestly, almost the only time I use Mavericks is to run Parallels 9 and Windows 7 and that legacy Windows app.
4) My original Mac is a mid-2007 MacBook is still running OS X 10.6.8. That's what the HP printer is connected to. I now use the MacBook for email or when I need to take a computer somewhere else. My MacBook is not capable of being upgraded to run Mavericks, and I wanted to at least know about the current version of OS X, even if I NEVER upgraded my main system.