You are quite right, this is the control board for the active thermal camera.
The MODE button in a cycle switches the operating modes when this is used as a standalone device.
PA9/PA10 - this is also correct here! In one case a 1.5Kohm pull-up resistor is needed, in the other case it is not needed, but this is a feature and inconvenience that I decided on consciously. That is, if necessary, I can put a jumper and thus control Pullup DP.
You recreated my settings exactly from the file. I also tried Internal clock and that also did not lead to anything.
Regarding the supposed cause and the start of the firmware.
In order to exclude such errors, I generated two firmware for stm32f103 and stm32f070.
And flashed them from address 0x0800000000 via ST-link and STM32CubeProgrammer.
For verification I erased the chip, checking that all memory is filled correctly
But only stm32f103 starts and the LED on PC13 confirms it. And yes, I also check for a signal from the crystal and only the stm32f103 has one.
I will not argue about the quality of st-link and DFU, but st-link, being an official tool, shows its reliability in other projects and even on the example of stm32f103 everything works perfectly. I should check the DFU too though.
I have removed R172 and put it back in place. This does not lead to anything. And of course I connected my board to Linux, because I just recently got a Windows computer and I can’t be sure that I will do everything right in Windows.
If you do, then you can do a DFU-UTIL load of the Klipper firmware and start working with it once you reset. I would recommend changing your config to use USB rather than serial as that will give you an immediate link to your host.
I would like to understand what I am doing wrong with stm32f070, because I am doing the same actions for both MCUs but only in one case I get the desired result.
If you have any suggestions, what could be wrong with the stm32f070, I will be glad to fulfill your recommendations.
I would be satisfied with one of the three LEDs glowing or the presence of a signal from the crystal, the rest will not cause problems 
I emphasize.
The operation of the hardware has been checked.
All actions for stm32f070 and stm32f103 are similar
The result have is only for stm32f103