I currently record from my server PC to my NAS which is connected to a gigabit switch, and have no issues to speak of (very occasional artifacts, but that's normal with my cable stream in general). I do have a network tuner, though, so some amount of network traffic is required in my case.
If you do a lot of recording or streaming at the same time you might eventually eat up a gigabit of bandwidth, but that's a lot of simulataneous streaming/recording. A Bluray maxes out around 50mb/s, and HD TV recordings are typically much lower bitrate than that. So even fencing off a few hundred megabits for overprovisioning, you'd need a lot of simultaneous streams to hit that limit. You'd probably hit a hard drive i/o bottleneck in the NAS (from all the thrashing) before you exhausted the Gigabit connection.
All that said, there's no hard and fast reason not to record to the SSD unless you don't have much space. It will cause a lot of writes to the SSD (as the recording directory is also the timeshifting directory, and time-shifting is mandatory while watching live). That matters because SSD's do have a limited number of writes, *but* most of the consumer anxiety about limited writes on SSD's has proven to be overblown, see e.g.
https://techreport.com/review/27909/the-ssd-endurance-experiment-theyre-all-dead . Based on that and other similar experiments, it looks like you're far more likely to see a modern SSD die of old age before it exceeds its write limit. The only real issue with writing to the SSD is that you have to actively manage the disk space (or script something), unless you have a terabyte SSD or something.
But Jim's right, a large local spinning drive is probably the easiest/safest solution.