The most likely scenario for a US president to order a nuclear strike would be if an incoming strike of nuclear nature has been detected. That decision has to be made and implemented within about 15 to 20 minutes, the time it takes for ICBM's detected in flight to reach their targets. It can be less in the case of submarine launched missiles, if the submarine has been able to approach the US coast, shortening the flight time.
Getting congress to agree on that in 15 to 20 minutes simply isn't practical. This is known as Launch On Warning - multiple ICBM's have been detected approaching the US in a ballistic profile that matches what would be expected of an incoming warhead.
Because the US is prepared to launch a full counterattack on a moment's notice, destroying the country that initiated the event, the US has not had to take this option. Neither do Russia, China, India, France, or the UK... the other nuclear/ICBM armed powers, because they have similar systems in place to immediately respond to a detected nuclear strike.
This is MAD - mutually assured destruction. While the acronym describes the theory quite well, it has prevented a major nuclear exchange for over 60 years.
However, in the absence of Launch On Warning, the military will want to verify any nuclear strike order. The officers tasked with initiating the launch will be held accountable for verifying that the order is legal, especially as it is very unusual and the consequences extremely severe. High ranking US military officers are not robots, especially those tasked with maintaining the nuclear arsenal. They are aware of both the circumstances that might justify a non LOW nuclear strike, and the consequences of such a nuclear strike.
The key here is the presence of an imminent threat. The only imminent threat that is considered justification for an immediate nuclear strike is the detection of an incoming strike... launch on warning. In that case, the president hasn't started a war, whomever launched the missiles has started the war. The president is simply responding.
It is true that a president can order a military strike of non nuclear nature, provided congress is notified of the action within 48 hours, and has 90 days to get congressional approval of the action or the funding gets cut off.
But, there is no possible outcome from launching a nuclear weapon that does not end with a full war. It's a red line that no president since Harry Truman has crossed, so any such order would be subject to extreme scrutiny by multiple people tasked with carrying the order out, because it is so unusual and the implications are so severe. This judgment would fall to the secretary of defense, who was approved by congress.
The president can fire the secretary of defense and put someone else in their position, but that takes time. Meanwhile, the vice president and either the majority of the executive cabinet or the majority of congress can declare the president unfit for office and remove them.
So it's not like a president can just wake up tomorrow and order a nuclear strike. In reality, the president's authority to order a nuclear strike is limited to a retaliation against a verified nuclear strike coming at the US.