While most of us keep a rotten rooster far too long the firs time, I can see your hesitation. Thing is, they tend to be opportunists, and the time you have your hands full, or your attention somewhere else, that is when they tend to give it a try. Many times people on here, will tell of him starting an attack from the back, to stop it when people face him. However, that is a short term fix. He will get braver and braver.
IMO rooster behavior is more like waves, it is not (at first) a 100% I hate you. That comes later. It is let my whoop on you, and then you can be my friend...until the next time I decide to whoop on you.
Alphas and Betas - can generally work it out...until the Beta decides to be the Alpha. They can go from nothing is wrong, just being chickens, to a full fledge fight. Some times multiple fights a day, sometimes to death.
So while I can see you are against culling them, do not take the chicken theory to the emergency room. Be especially careful if you have small children as they will attack them first, and often times at the head level.
Good luck, but be aware, you are describing some very poor traits, traits that I would expect to get much worse. I have had chickens for decades.
Adding more hens really does not fix the problem of too many roos. Chickens really need a lot of space, and adding more hens drastically reduces that. You would need a much bigger coop and run. Adding chicks for future number of hens does not fix the too many roosters NOW problem.
Mrs K