The upside is that it's very pleasant to program compared with existing APIs, and is portable. Even when the GPU doesn't quite meet the requirements, it's possible to polyfill compute shaders with SwiftShader. As this standard becomes more mature, it will be reasonable to expect that it will be correct and performant pretty much everywhere.
Note that both web and non-web implementations are viable.
The biggest drawback is that the "MVP" covers the basics of compute shaders but not the advanced features. It's roughly equivalent to DX11 / Shader Model 5, which is of course good for compatibility on pre-Windows 10 devices. It's missing subgroup operations, scalar types other than 32 bits, an explicit memory model, and some other goodies. This potentially leaves quite a bit of performance on the table, but that will depend on the exact workload. For example, machine learning inference workloads will probably suffer considerably due to the lack of f16. Thus, if you want to ship something based on compute shaders today, it makes sense to build your own portability layer on top of existing APIs such as Vulkan, DX12, and Metal.
I do think WebGPU is an excellent on-ramp to learning compute shader programming, as it will get stronger over time, and concepts will transfer to other APIs as listed above.
Watch my Twitter feed for more depth on the points I listed above.