Hi, Bugsy. I've been a software developer (or computer programmer as it was called back then) for my entire life, I have over two decades of professional experience and over three decades of coding experience, so I can totally relate to what you are feeling.
Just how in the world does one keep up with it all?
A: You don't. Keep happily focused on what you like / want to do best, ignoring all the rest.
If I don't keep up, will I still be taken seriously as a developer?
A: Yes. When you build up 10 years of expertise in technology X you will be taken very seriously for positions concerning technology X.
Is there a place in the dev community for those who are marketers first, and developers second?
A: Absolutely. Being able to market yourself is a skill as important as (if not more important) than any other skill. My advice is to not abuse it though, you'll sleep much better by selling skills that you actually possess.
Is becoming full stack out of the question?
A: Nope, full-stack positions keep popping up all the time. Being able to put a full solution together is highly praised today.
And alas, if I don't use React, am I still a developer?
Yes and personally I think you're probably a good one for that exact reason. (Sorry, I could not resist it.) Try to ask yourself that same question in a couple years, but replacing "React" with "The Next Tech Fad". Think of all those buzz words you are probably glad you did not waste your time with when you look at your skill set today. I could name a few, but I'm sure you can easily remember those that used to be a thing yesterday and now people don't even remember about.
First things first: tech fads come and go, some stay longer, some make more noise, some don't. Also, I am sure I have seen at least a couple interesting job opportunities for PHP developers recently. Not to mention I see some FORTRAN and Cobol positions available from time to time, but let's not get to extreme-legacy tech.
My point is: you don't have to catch up with everything that gets in the ever-increasing list of tech buzz words. One can always grow, be it in breadth or depth. Usually depth is harder to achieve and a lot valued in the job market. I hope you can enjoy the simple math in this example:
Billy has 2 years of expertise in Python and 2 years of expertise in Ruby;
Jimmy has 4 years of expertise in Python;
Company X needs a developer with at least 3 years of expertise in Python.
Which candidate do you think has a higher chance of getting the position at Company X?
Also, when I mention depth I'm not just talking about having decades of expertise in a particular programming language, but adding up to your core computer scientist skill set. No matter which language you write in, the core principles are always there. If you understand them well, you can easily navigate between and excel in any kind of programming language.
That being said, it doesn't hurt to sometimes get out of your comfort zone and try something different here and there. Specially when you notice some specific tech is getting the right attention building things you are interested in doing yourself.
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