I don't know about "irrelevant", just not applicable to the President's powers to wage a war. There are a number of provisions of law that are only activated under a declared war(such as enhanced punishments, and suspensions of rights), and I don't think they've been tested.
Actually, that's not quite right. The suspension of rights is a distinct act on Congress' part... it doesn't automatically attach to any warfighting. The "president's power to wage war" is entirely constrained by Congress' Constitutional powers. It's just that they delegated many of those decisions to the executive in recent years (war powers act, etc).