I ran a simulation using the plate and heat sinks with convection and emission:
I really doubt this is correct though ... I have to investigate a bit more (need a more detailed model of the LEDs; right now they are 13mm aluminum cylinders) and remove the heat sinks to see if the temperature goes up or if I messed something up.
Again, just based on pure guess I'd say a large plate and 4 heat sinks should be enough cooling and much easier than stripping paint from heat sinks and welding them together. I'll post another update when I make sure the simulation results are correct
Edit: I figured out what I did wrong. For heat load, I used 10 watts (14 watts LED at 30% efficiency) and selected all the LEDs thinking it applies 10 watts per LED, but fusion meant total power. After increasing it to 160 watts (10*16) this is what I get:
This is assuming perfect transfer between the LEDs and plate to heat sinks (no thermal interface material / assume the entire thing is machined from a solid piece)
4 heat sinks were definitely a no-go. Funny enough increasing them to 9 didn't improve things by a huge margin:
The inner LEDs are just on the edge but the outermost LEDs seem to be fine. Now if we add the thermal paste inefficiencies back into the equation, even 9 heat sinks won't be enough.
Edit 2: I have some bad news for you: after playing around with the simulation and using BOTH the heat spreader plate AND 16 heat sinks, the LEDs are at 50c with an ambient of 25c (25c delta). Again this is assuming no thermal paste (perfect transfer) so in reality, they'll be right around the 60C mark. Unless you want to run the LEDs at a lower power level, this design is very impractical on its own (ignoring RGB). If you want, you can send me your fusion 360 account so I can add you to the project to play around with the model yourself (everything is parametric and easy to change).
I'm not sure how they measured the 2K/W (maybe with a decent amount of airflow?), but 16 heat sinks will definitely not be enough on their own. Increasing the convection value from 10 to 100 (aka using a fan) does improve things a lot but you said you don't want to do that.
As a side note: heat sinks are extruded aluminum so you may not be able to easily find them wider than 100mm, but I'm sure you should be able to get longer ones (for example 100x500mm). Try contacting the seller to see if they can send you longer pieces (without cutting them down to 100mm).
Edit 3: using the spreader plate and 4 heat sinks + airflow (fans) results in a max temp of 46c which is only 10c warmer than using 16 heat sinks! Increasing the plate thickness from 3 to 5 drops temps by another 3 or 4 degrees! This should tell you that the heat sinks you chose are a very bad choice for passive cooling and when using fans, just 4 should be enough.
As you can see the innermost LEDs get a bit hotter than the outermost LEDs so you might want to run them at a bit lower power which is easy as you have full control over each individual LED from the driver board (it's a good idea to actually check the temperatures using a thermocouple after you build the whole thing).
You can also watch this video :
where they add a "massive" (compared to what's already in the steam deck) heat sink, but after about 30 mins, it reaches the same temperature as it did without the heat sink! but adding a small fan makes a huge difference in the end so they decide to keep the fan on there.