There seems to be a component with the third one you've marked, see the back of this card on TRW. It's not the exact same core, but PCB wise it should be the same.
I believe it is used for some kind of Vref, although I don't think it being missing would be the case for no POST on mainboards. Do you have a POST analyzer card to check if any POST codes are executed?

I also thought that the missing third element shouldn't interfere with POST. Unfortunately, there's no POST card.
I haven't tried flashing the BIOS on the card, but as far as I understand, its damage or even its absence shouldn't affect POST?

It's not for VREF. The PCB you have is of the 9700 Series and they slapped a R350/R360 onto it, as it is pin compatible and sold it as a 9800SE.

This part down there is from the hijack circuit, apparently the R300 needed this for coldboot issues, it directly interferes with AGP signals.

Look for U106 on the page of the "hijack circuit".

Thanks, u106 is LMV7219M7. I found where to order it, but the delivery is very long - more than a month. Could the lack of POST be due to the absence of u106? And is it possible to temporarily replace it with something to check it, I saw somewhere that they soldered a resistor between pins 1 and 3.

Can you check if the pads really look like they've been populated before? Perhaps a close up picture?

If the card uses it, the part is meant the generate a TTL compatible DEV_SEL (device select) signal for the following logic. The one coming from the AGP slot is not TTL level and thus they're using a comparator to compare it to the AGP reference voltage. The resulting signal goes into the rest of the hijack circuit (as they call it).

You could use a TLV3511DCKR instead, if it's easier to get for you.

The solder on several of the u106 pads was scratched, so I removed it completely using a braid. There seemed to be traces of broken legs, but I'm not sure. In all the photos of the 9500 and 9700 boards, the u106 is installed. I only found one photo without it, but there are no resistors or capacitors around u106, while my board has them. Going back to the beginning, is this a false trail, or could it be causing POST issues, and should I order it and wait a month? TLV3511DCKR is even more difficult to get in my country.

MaRCer wrote on Today, 15:32:

The solder on several of the u106 pads was scratched, so I removed it completely using a braid. There seemed to be traces of broken legs, but I'm not sure. In all the photos of the 9500 and 9700 boards, the u106 is installed. I only found one photo without it, but there are no resistors or capacitors around u106, while my board has them. Going back to the beginning, is this a false trail, or could it be causing POST issues, and should I order it and wait a month? TLV3511DCKR is even more difficult to get in my country.

Yes, it can 100% be cause of the non POST. Where are you located, if I may ask.

Russland

Aye, gotcha.

Later revisions of 9700s didn't use the hijack circuit anymore. 9800s didn't do it in the first place. It seems to me your board is just really a recycled early 9700 PCB that was re-purposed to get the storage shelves empty.

If you want to try something out and are comfortable with it, remove K1 and K2 and bridge their pin 4-5, effectively connecting the GPU normally to the AGP port. (blue in the picture are the pins to bridge)

Schematic part

Great, I'll try it. How do I properly remove the heatsink from the GPU, is it glued or just stuck due to dried thermal paste?

They used a very sticky thermal pad/compound.

My go for it is: remove the fan, then heat up the heatsink with hot air and carefully twist it.