Can Ultimate Abilities Have Cool Downs?

I feel like if someone uses an ultimate ability it shouldn't be back up or almost back up when they get back to lane few examples is

his ult is SUPER good at wining close fights and diving because he goes into undying rage and can't well...die so you feel like since he used that you have a window of opportunity to fight him and kill him and this is really the only time you can fight him because if he cant die you cant kill him but the CD is so low he has it up every time making it nearly impossible to fight him 1v1

His ult is good for engaging and defensive so if he uses it on you youll probably be too low to actually fight him and if you go in on him hes gonna place it on himself and just end up in you dying. The same deal is you would think after he has use his ult once you get back to lane you have a window of opportunity but he has it back again and if he doesn't its close to being back

Not only is he hard to fight in general but even if the fight is close he will win because of his ult and he will have it back for the next fight

I just dont understand why this is the way it is if you play anyone remotely weaker in lane you will always lose and I'm fine with champions being lane bullies thats not my issue my issue is that the window of opportunity is so damn small you cant do anything with it.

Also please note that I'm not saying these champions are broken I'm just saying their ults are on a really low CD