If what I understood from the quests thus far is correct, then Shadow's Edge is the weapon you will create from the Primordial Saronite, Light's Vengeance (Arthas' weapon), etc. That, in my opinion, seems to be the hardest part from a resources point of view. After that, your only bottleneck is your rate of progression through the raid. You will eventually be able to soak Sindragosa's Frost Breaths, collect the acids, and do what you must with the Queen encounter. Then, it's just a matter of time before you are able to collect the required number of Frost Shards. Therefore, I don't think "everyone" will be able to get Shadow's Edge, at least early on because guilds will most likely delegate one person to receive all the resources - if not the Primordial Saronite because you have the gold/badges to buy them yourself, then at least the Shards off the boss.
For this reason, whoever is destined to receive the legendary axe should, economically, get Shadow's Edge. Afterwards, it is just a matter of a time before Shadowmourne is fully forged.