Each NVMe drive uses 4x PCIe slots. 3x NVMe drives is 12x PCIe lanes, I'm not even sure if an AM4 motherboard would support so many NVMe SSDs.
I'm only aware of Threadripper motherboards that have that many NVMe slots. Maybe with port-bifurcation and a Hyper M.2 x16 card you might get there but... I've never done something with that many NVMe slots.
You should benchmark your system and try to determine the actual bandwidth-requirements of the 7-man TB. SATA-SSDs are a lot cheaper, and hard-drives are way, way, way cheaper. Serving so many I/O slots is going to be expensive.
I don't actually know any LeelaZero performance characteristics. So I'll defer to others.GPU: I might be looking at 1 or 2 X RTX 2070 Super; or is this a waste of time and not worth bothering about because I really need 2 X RTX 2080ti to be competitive in lc0?
* Case: This is a deeply personal choice. Find where you want to put the computer. Measure out your room, and determine the size. In general, bigger-cases are easier to work with (more room for your hands to fit inside as you plug cables or move stuff around). If you don't have any preference, I'd suggest any $100+ Full Tower case as a beginner build. If you have space constraints (fit under a desk or whatever), you'll have to measure it out and pick it out yourself.What about RAM, case, and cooler recommendations?
* Power Supply: Hard to say how big you need yet. Once we nail down the components, we might be able to pick out a PSU.
* RAM: Chess likes lots of RAM, but I don't think chess-engines care very much about the speed of RAM. Video games care a lot about RAM Bandwidth and latency timings however. I'd suggest a set of CAS16 1.35V 3200MHz DDR4 sticks as a starting point, unless you knew that you needed better specs than that (ex: Lower latency or higher bandwidth). Be sure to test XMP settings and run Memtest86 to verify your RAM is free of errors.
* Cooler: A high-end tower-cooler for $90 is sufficient for any CPU 200W TDP and less, while higher end air-coolers can go up to 250W dissipation. I don't like liquid-coolers at all, they're too risky. Air is simpler to setup. There is a convenience factor for liquid-cooling: you can move the radiator around the case and optimize your air-flow. But the added complications (case-compatibility, leakage worries, pump-unreliability, etc. etc.) makes me stay away from liquid.