In a situation without transcoder, RAM and CPU are definitely important factors too. It’s just that, in most cases, the network interface is the bottleneck (exceptions are situations where you have a disproportion, like a single-core CPU, or 2GB RAM alongside a 10Gbps NIC)
In a situation with transcoding enabled, the case becomes different. Transcoding is a very CPU-intensive process; each incoming feed must be decoded, and then re-encoded to the different new bitrates (in your case, 720p, 480p and 360p). That means that the bottleneck now becomes the CPU, and the number of streams you can serve from a server simultaneously is much more limited. Don’t confuse this with client connections though: one incoming stream can still have 1000s of clients. It’s not the number of client connections that gets limited, but the number of concurrent incoming feeds.
There are several options to build a successful business case when you need transcoding. On option is to consider either an NVidia GPU in your server, or use multiple smaller servers with CPUs that have Intel QuickSync. There are also live and VOD transcoding services available online; some built-in with a complete streaming service (e.g. Wowza Cloud), some as separate service.
If you’re interested in Live transcoding as a separate service, drop me an email: email@example.com