This is incomplete. The treaty-making power is shared with the Senate, which is required to "advise and consent" by a 2/3 majority to any treaty.
The "advise" portion has never really been implemented, but the consent portion still works. So the president cannot make peace without 2/3 of the Senate consenting. Though the Congress also does not really have the power to force the C in C to wage a war of which he disapproves.
The power to make war is also not exclusively legislative. Depending on what you define as "war," the US has been involved in some dozens of wars. The Congress has formally declared war only five times, most recently in 1941.
The control of the Congress over presidential war-making is much more related to the power of the purse than to the power to declare war, which we don't do anymore, though congressional authorizations to use force, as in the Iraq War, are IMO the functional equivalent.
As I said, the war and peace making powers are both shared between the legislative and executive.
As I've given the Constitutional Articles and clauses for my assertion, please be kind enough to provide sources for your claim.
I figure the ‘advise’ part worked with the league of nations, whereby reservations were developed, and the league of nations treaty would not be passed without the reservations. When the reservations were not included, the president rejected the advice, and so the treaty was not passed.