There is most likely a backplate behind the MOBO that the screws were screwed on. You have to remove that backplate and that possibly means that you have to completely remove the MOBO and get the backplate off. When you have done so, then you can install push-pin CPU cooler.
Taken many of those apart. They are standoffs screwed into the motherboard tray. Some are even pressed into the the tray. It may be easier to get a new case then to play around with trying to get the new cooler to fit.
Before removing the threaded holes, check that the heatsink will fit properly since the heatsink fitment is socket specific.
The screws holes can normally be removed by pushing firmly, they are usually held in place by friction only. I'll suggest pulling the black plastic caps off first (which may have some adhesive on them) then pushing the threaded holes out.
You will need to put new thermal compound on the CPU before installing the 'new' heatsink
Hope it helps
Removing Dell motherboard back plate. I have done it and will provide some guidance based on my experience with Inspiron 570.
First of all this is how your back plate looks approximately (last reminder - all shown here are Inspiron 570 parts so you might have to make some adjustments)
Here is the part which need to be removed and instructions, I don't think there is something on youtube, since I made those instructions and pictures are mine including wasted Vons card...
To remove backplate you will need credit card type plastic, any plastic card will do, no metal to touch motherboard! I do my recommendations in metric system, I don't like inches...
Here is backplate. I used as you can see 4 peaces of plastic folded in half, approximate size (width) of already folded plastic is about 1-1.5 centimeters wide and 4-5 centimeters long. First you have to use plastic card (library card, credit card or similar) to lift backplate up a bit to insert those plastic assistants. Backplate is glued to motherboard! As soon as you inserted those FOLDED plastics, you can stick regular, flat screwdriver into the plastic folds and now you can use more force to lift backplate off motherboard, there is no rush here, you might need about 1 hour of monkey job. Lift a bit on 1 side, advance plastic closer to center, move to another corner, and so on. MAKE SURE THAT NO METAL TOUCHES MOTHERBOARD!
The glue, sticky paper located on the outer sides of backplate square as you can see in next picture.
As soon as backplate is out, you have to carefully remove this sticky paper, here you can use only you fingers and nails - no metal here as you can easily scratch motherboard, improvise.
The interesting part is your push down pins - would they fit into the holes?
Can you return your CPU cooler? Or it is too late?
There is another way, which could work on the existing backplate, but you would have to improvise - remove push down pins and substitute them with screws.