Typing "INT" (for Integer, as you recommended above) yields "Language not available." Typing "FP" does nothing, just gives me a new prompt.
That would be normal, again, for a machine with no language card installed.
(History lesson:
As I mentioned earlier, there are two mostly incompatible BASICs for Apple IIs; the original "Integer BASIC" as written by Steve Wozniak and descended from the BASIC he wrote for the Apple I, which is fast but syntactically limited, and "AppleSoft Basic", a highly customized version of Microsoft's 6502 BASIC. (IE, it's a cousin of, say, Commodore BASIC, in that it's descended from the same codebase.) The motherboard of an Apple II (pre-IIe and later) has sockets that can hold *one* of them; to run programs written for the other BASIC the original solution Apple offered was a ROM card that stuck in the slot the language card goes in and actually had the ROMs for the other BASIC on it. However, around the time the IIplus became a separate "model" (rather than just a "II with AppleSoft installed) Apple also begin offering the language PASCAL for the II and since PASCAL wanted as much RAM as possible *and* running Pascal the memory space occupied by the BASIC ROMs was wasted anyway Apple came up with the "Language Card" with RAM instead of ROM on it. The advantage of having one over the ROM card is it lets PASCAL or machine language programs use almost 64K of RAM instead of 48K, *and* it can also double for the ROM card by loading the desired BASIC into it, so it quickly became the STANDARD configuration for a ][plus...
But, still, as you can see by the prompt you got when you booted DOS 3.3 it is still technically a valid configuration to *not* have one. It just means that, again, since you have "FP", aka, Applesoft, installed on your motherboard and no ROM card plugged in in place of the language card "INT" is indeed UNAVAILABLE.)
I think we're *pretty* convinced at this point that your system and disk drive basically work minus the language card; if you wanted to you could dig through the disk archives and look for some disks of AppleSoft programs or machine-language programs compatible with a 48k Plus and exercise it, but since it passes diagnostics I'm inclined to think you're good. So... the question is what's wrong with the card. I assume that if you move the chip back over to it and reinstall it you're again not able to boot; that is of course a pity because it would be great if we could run that diagnostic on it. Lacking that...
The *simplest* problem to diagnose would be the language card having some bad RAM on it. If you are feeling adventurous you could *very carefully* remove all the 4116s from the language card, assuming they're socketed, substitute them for one row of RAM chips on the motherboard, and see if you can still boot the diagnostic disk and run the memory test. If the test fails then you've found your culprit.
Before doing that, though, I would suggest *very* carefully cleaning the contacts for the cable that goes from the language card to the IC socket using some alcohol or contact cleaner, examining the socket the cable jumpers into to make sure it doesn't look corroded or worn, and maybe gently polishing up the card edge contacts with an eraser. I *think* the refresh and CAS/RAS signals for the RAM on the card are carried over to it by that cable and a bad connection could be making them "noisy" enough to cause errors. Worth a try, anyway.