It _is_ terribly difficult to manage unmotivated people, but most people only end up unmotivated because of poor management. And it's usually cheaper and ultimately better to motivate a proved achiever who has lower motivation than to find a replacement.
A manager should always start by assuming that a motivation problem is something he can address, and work with the employee to figure out what the real reason is and see if they can make the necessary changes. But sometimes, the employee and organization just aren't a good fit anymore, and it's better for both of them if they part ways and find new situations that are a good fit.
I've encountered only five managers who fully exemplify this, but they had an outsize role in keeping good people who others might not have been able to keep, keeping them productive (at least enough of the time to satisfy their work commitments while they dealt with other issues), and attracting more good people (because of word of mouth from the existing good people). It seems the key here was to have the conversations when they were needed. The advice here "Make your job easier and don’t." doesn't correlate with these great managers at all. At least two of them to my knowledge have given reports the advice (and assistance) to find opportunities elsewhere.
In smaller orgs with no HR, it's probably easier, but larger companies have a specific process.
Though to your point, I would not consider myself someones manager unless I had the option to hire, fire, or promote them.
In addition, don't just give up. Keep talking and helping out. It could be advice on how to stay motivated or at least not demotivated, carving out space for interesting projects or developing new skills, or even helping that person find a new opportunity outside the company.
Dealing with unmotivated unproductive people you have limited ability to actually manage is beyond my experience.
Motivation is the responsibility of the employee. As a manager, if your performance is faltering and it's because you're not motivated, I will tell you clearly my perspective, and ask if that is the case. However it's on you take corrective action. I will do everything I can to help you, but it's your responsibility, and not fair to the other team members who are motivated if
1. I am focusing my energy dealing with issues caused by your lack of motivation.
2. They're working extra to ensure success when a team member is unmotivated.
Whether or not that has to do with their motivation is unclear.