Offering up my experiences and viewpoint as an older developer.
At 45, I've been professionally coding for just over 20 years, so I've been able to work with the majority of technologies that have been available during that time span. Just for a frame of reference, I've worked in consulting, manufacturing, owned my own business, for health care companies, and most recently, a NLP "Big Data" (quotes intended for lack of a better term) company that is all OSS and cloud infrastructure.
First off, I don't feel like an older developer, as I still have the same passion for building software that I did when I was 24, but how I approach it is different. I have no real interest into going into a management position as it would rob me of my main passion in life. However, by being heavily involved in the local developer community, I've been able to help many people who are new to the field, and companies who need technical help at an architectural level, while not distancing myself from actual development as my primary job.
To address your questions specifically...
If pay and influence are your primary concern then traditional management is obviously one route, but I offer up the alternative, only because it has been my experience. That alternative is being involved in helping people in the community. Speaking at conferences to share your skills and experiences, attending hackathons to help people build interesting projects and meet even more interesting people, working with teachers/students at local schools, or any other types of event where you share the expertise you've gained over the years can help serve the same goals. You gain exposure, so more people will want to work with you which may easily lead to increased pay and influence.
As for there being a plateau in pay, I can't answer that, but I can definitely attest to there not being a drop in pay as long as you don't let your skills atrophy. There is a huge need in the technology world for experienced developers who can guide product development either individually or as part of leading a team.
"Are older devs not looking for jobs..." I think this is a faulty assumption. There are plenty of older developers looking for new opportunities, but I cannot deny that there is a significant segment of the workforce that does favor stability. Again, in my experience most companies that I talk to are looking for a mix of the 5+ and 20+ crowd. Only the most naive would hire a team of developers in their 20's and expect a robust, maintainable, sustainable solution... unless, of course, that's not the goal.
As for the specialist vs. generalist question, I think the distinction is irrelevant and either route would be fruitful. I might give a slight advantage to generalist because then your personal opportunities increase, but there's room in the industry for both, as they are both valuable.
So currently I'm coding and mentoring at my job. I know many other developers in my area that are approximately the same age doing either pure development, a mixture of development and being a team lead, or they've moved on to management.
I think the point is that the path of a passionate developer is not set in stone. You don't have to do any one particular thing in order to advance a career. The important thing is to figure out what's really important for you.
Do you care about remaining a developer?
Do you care about gaining influence and/or getting a high salary?
Do you care about helping your community?
Do you care about educating or mentoring others?
There's no one path.