It won't make streaming slower unless you have multiple clients streaming, as in several.
I find the network to be the bottleneck - my gigabit network connection saturates with 3 streams, and I'm using a conventional hard drive for my media (OS is on an M2 drive). This doesn't seem to affect the video quality though.
Frankly SSD is overrated for common stuff, there are other bottlenecks that usually hit us first, such as network or processing.
As you build out, make sure you consider backup in your costs, don't spend your money just on storage.
Also, since you have a mini there may only be room for one drive internally and almost no cooling. Larger drives will have issues with heat.