If you know who Daenerys Targaryen is, you do not have "no time." You have priorities which included "I wish to know who Daenerys Targaryen is." No judgement! I do, too! But if you know who Daenerys Targaryen is and you're dissatisfied with your career growth, you should attempt to consciously be aware of the fact that you can choose different things in the future.
It takes minutes to set up a web presence as a software developer. It takes 1~3 hours to stick a flag in the Internet about one topic. Having one flag stuck in the Internet beats zero flags, by a lot. You can keep doing this as you have time available. (Think less "submit bug fixes to OSS projects" and more "write an essay which shall endure regarding a topic which is professionally relevant to you." To pick one of the hundreds of things in this genre produced by an HN member, see https://www.chrisstucchio.com/blog/2012/hft_apology.html where HN's own yummyfajitas explains, in a useful and citable fashion, how order books work. You almost certainly know one topic well enough to post something which clears that bar of utility on any given day.)
You could also, I don't know, invent Rails, but you do not have to invent Rails to get career benefits out of working publicly.
After you have a flag stuck in the Internet, there are a variety of ways to get value out of it.
The simplest one is using it as a conversation piece when you're cold-opening conversations with folks who are relevant to your career. For example, if you've decided to stick a flag into (picking a random example) ActiveRecord optimization, and you're attempting to talk your way into a job interview with a Rails shop, "I see you're a Rails shop. I would like to talk to you about potential opportunities at your company." is _substantially_ improved by "Hiya, I see you're a Rails shop. I wrote a detailed guide to ActiveRecord optimization (see here). I would like to talk to you about potential opportunities at your company."
Another one -- also not hard! -- is to put a call-to-action at the bottom of the essay saying "If you're reading this, it is probably because you are interested in ActiveRecord optimization. So am I! If you've got hard problems with ActiveRecord optimization I would love to help you solve them. We should chat; here's my email. "