CHEAPO Push pin solutions do not and will not EVER work.

Dell's backplate is proprietary as is their fan and fan wiring.

The solution is to replace the black plastic HS/FAN with a Different Design that is also from dell but is 100 percent METAL construction.OPTIPLEX 960 / XPS 435MT CPU HEATSINK COOLING FAN P041K CN-0P041K

The P281K that failed is not very good.

https://www.amazon.com/Optiplex-Heatsink-Cooling-P041K-0P041K/dp/B01666A8GC/

On such an old machine, I think I would have cleaned up the cooler of dust, replaced the thermal paste properly, reinstalled flat and properly ... and called it fixed.

If the cooler's fan was loud, (and hard to source a better new/used one) a drop of silicone oil (or 30w motor oil) would likely have silenced it.

Yes. The plastic one gets brittle and breaks.  The metal one does not fail.

Dell Part Number: P041K 0P041K CN-0P041K
Compatible Models: Dell OptiPlex 960 Studio XPS 435MT MiniTower

https://www.dell.com/support/home/us/en/04/product-support/product/studio-xps-435/manuals

https://www.amazon.com/Dell-Heatsink-Assembly-LGA775-Optiplex/dp/B004YWSBII

Well I know the quarks with push pin coolers but they will fit in the case and work. You do have to remove the back plate. I went & got one by Rosewill I believe. And it works fine. Temps are fine with it. The one that comes with the Dell is cheap. The whole plastic breaks off at the screw point. 

Push pin at times 1 pin acts like it don't wanna hold right. But taking your time and even doing it with the board out of the case will make it easier for you to get it to hold. 

Metal will of course always better. But not necessarily needed. 

The metal versions need to be replaced 1 time and never need replacing again.  The Black plastic ones become brittle and crack and break requiring replacement as they cannot be fixed once broken.

Thank you so much is there another heatsink I can use with the same screws  to attach it ? your post was extremely helpful , thanks so much for your time and effort

So I bought a P041K CN-0P041K, but the hold down screws are not the same.  The ones on the P041K are thinner and a different thread than the ones on the P281K.  Any ideas on how I deal with this issue?

Thx\Chuckj

Re: So I bought a P041K CN-0P041K, but the hold down screws are not the same.  The ones on the P041K are thinner and a different thread than the ones on the P281K.  Any ideas on how I deal with this issue?

This is a familiar question.
P041K is for 435MT.  Lga1366 socket.  There are tapped holes built around the socket from a permanently attached back plate which is integrated with the motherboard.  Heatsink screws are directed mounted onto motherboard.  These are typically fine thread screws.

P281K is for 435T/9000/9100.  lga1366 socket. In 435T Dell uses a black backplate JU616. This back plate is mounted on the floor of 435T case.  It is not fastened to motherboard.  When you remove mobo you can see this plate still sitting there.  The holes of this plate exactly matches the cpu socket on motherboard in location.

435T motherboard has clear untapped holes around socket and heatsink screws go through these holes and fasten to the backplate.  These are coarse threaded screws because JU616 is a piece of hard plastic (you cannot tap fine threaded holes in plastic). 

Therefore heatsink for 435MT and 435T are completely incompatible.

You should return P041K because it is incompatible with 435T.

435MT

P041k for 435MT only

435T

JU616

P281k for 435T/9000/9100