Why no one was/is alarmed enough to build a peer navy to the USN ? Because despite any hard feelings these countries may have for the US, countries the world over know that the presence of the USN, particularly a carrier fleet, is a good thing for any troubled region. You may not like US as the self appointed "world's policeman" but you sure are damn glad that we are there when there are problems in your backyard. I do not mean 'you' as in China. So in a manner of perspective, the USN is the world's navy.
So who is/are going to spend vital national resources just to try to be a peer to the USN ? Of all the major powers in the world, only Russia and China will attempt and not likely the Russians, not even in our great grandchildren's lifetime. That leave China as the most likely aspirant challenger to the USN.
I said 'aspirant'...
(of a person) having ambitions to achieve something, typically to follow a particular career.
Ambition is not a guarantor of success. The driver, may be, but not guarantor. People casually say 'decades' as if time is on China's side that the PLAN, or whoever is the designated successor, will be a peer to the USN, and that the US and her navy will remain static for China's benefits. But the reality is that even if the American economy grows half as much as China's over the next hundred yrs, a wildly optimistic projection for China, the USN will continue to maintain a decisive technological edge over the PLAN in that hundred yrs time span.
If in 'a true opponent' you mean the POSSIBILITY of anyone else besides China, you are engaging in wishful thinking. By the time the PLAN's sailors mastered carrier fleet operations, not just in training, but in deployment to any part of the world under any sea condition to face uncertain challenges that taxes their intellect, endurance in terms of human and hardware, stresses of environmental and political natures, and return to home ports, the USN will have relegated the aircraft carrier to secondary threat status.
Despite being Air Force, I am willing to go on a limb and say that no 'true opponent' will rise to be a peer to the USN for at least one hundred yrs. An opponent to the USN implies that there are enough countries in the world hostile to the US that many of them will attempt to have their navies be a peer to the USN. Are you seeing that today, not just in terms of ideological/political differences but couple them with sufficient national resources to do so ? No, there are none, and believe it or not, no one want to. No one is throwing up his hands and give up. They do not believe an American President is going to wake up one day, throw a dart at the world map, and send a USN carrier fleet to that country's shore and invade for oil and women. No one throw up his hands and give up because they believe the USN is the world's navy, just like no one is going to petition for an alternative police force when the current shire-reeve is doing just fine.
But look at the alarm the PLAN's modernization and aggressive behaviors raises among China's neighbors today.