No it will not be too little too late. If you were in the U.S. Navy's shoes, you keep your distance right just like you said. Small attack craft have limited range so they won't go far out into the ocean. So they are limited to staying in the Straits unless they decided to go suicidal charge out into the sea. De-mining would not be the problem because they are not going to rush in. You know that for a fact because of anti ship missile threats. You clear out any SAMs and hunt down any coastal batteries.
My point was more along the lines of the following:
Closing of the strait via mines and subs
would force the U.S Navy and Helicopters to conduct anti-mining operations and naval operations. The issue as you said, is that the coast would need to be cleared. This is not an operation that would last a short while, therefore the point of my argument is that, closing the strait would force the U.S to operate in the strait
before air defense and coastal batteries are cleared, in addition to fast attack craft. This is due to the economic fallout as a result of the close. It's not acceptable for U.S or anyone to have that strait closed for too long.
You also have to keep in mind, my good friend, is that the U.S 5th HQ in Bahrain and the largest army base in Qatar will defenseless as Aegis equiped naval vessels will be no where near to protect them. A barrage of missiles would make quick work for the Patriot batteries there.
I think this is the likely reality, where the U.S is stuck between a rock and a hard place. Sacrifice the bases?
OR high-risk operation to protect the bases but with higher casualty rates by putting vessels in the front lines.
