the biggest "problem" with DSO1200 is amount of parts and tools necessary to do deep check. The DMM is using own supply, one can test it with separate power source and check the data coming from (it is FS9922). Near DMM chip there is an MCU, if i'm not wrong it was an STM something. There is JTAG port near that MCU, so esy to check if it's working. Then there is another MCU (using 24MHz crystal, 24ssop) - markings removed of course, but this is usb device so easy to check. There is an FPGAs (with heatsink). It seems to run NIOS i think, at least it does have separate SRAM (2x 256Kx8) for what so ever reason, and it does use shared SDRAM. There is DSP or ARM in TQFP176, using that shared SDRAM as well (64Mb SDRAM). When one look exactly, there are some JTAG ports here and there so not a big deal to check function of all that ICs (it is good that Hantek likes to have JTAG's easy accessible).
All power rails can be easily checked, typical Hantek configuration are bunch of TI ICs, with some 0R at the end of each supply (so one can easily remove them and check supply with no load).
If all that things are ok and still nothing on display, i would say check that NOR memory. All other parts, even that one FPGA are not really important, one will get still something on screen (as long they not shorting rails or bus)