How can I make a cheaper, more modern UHF RFID reader board?
The only UHF RFID Reader IC makers I know of are AMS, EM Micro, NXP and Impinj. (AMS recently sold RFID assets to ST Micro)  

To use any of these ICs, you have to design RF hardware and then invest in thousands of lines of code to configure the IC, read the tags and output the data.

AMS has done the best job of publishing reference design boards and code, which is exactly why you are seeing designs using their AS3992 IC.

I therefore conclude that it's unlikely that you will come up with a cheaper board in a reasonable amount of time.  

In fact, I would recommend that you seriously consider at least starting with a good reference reader such as the Thingmagic readers and OEM modules. You will probably find an AS3992 IC inside!