Here's what I've seen succeed from my experience.
0) Be comfortable with OR methods. I'm assuming this is a given for anyone who finds this stack exchange.
1) Know your markets. Some aspects of business need OR more than others. Operations management has had a lot in my experience (surprise). One buddy of mine took his clients from consulting and started his own practice doing procurement and network design. For the more technically inclined, I would think the same could happen if you sniffed around for work in planning, logistics, and scheduling. As you noted consulting firms play in this space because they're great at selling themselves. You can be too. Consulting firms can't match your rate if you're independent, and you would be amazed by the number of small to mid market firms who pay people full time to solve by hand the problems I mentioned above. Get out of your comfort zone and start going to networking events and see what people react to. Also, get on sites like upwork. My other most successful friend in OR consulting brought in all his gigs that way.
2) Kill your delivery. This could mean either being comfortable with financial analysis and powerpoint for more strategic projects or being a solid software developer for those more tactical. I think you should be good enough at both of those that you can get a job doing them alone. On the more tactical side of things, this is why you see software companies working in this space too because they're awesome at making OR methods super simple to use. If you're a decent developer though, you can deliver 80% of the value of a software company's product at 20% of the time and cost (and use some finance skills to prove it). I've seen people do this by writing their math models with an ORM (i.e. TicDat or SqlAlchemy) so you quickly get something with data constraints and few changes between dev and production environments. Additionally some background in web development can go a long way here. An API or a GUI will make it as easy as possible for someone else to consume your work.
3) Don't be afraid to jump. No one I've met doing this had a "right time" to start working independently. They worked contracts at night; they used their vacation to network; they left their full time gigs not knowing what their next contract would be 6 months from then. Worst case, you fail, you're out a little money, but you've gained a ton of experience. I think there are plenty of software companies and consulting firms who would hire you just on the story that you tried this, and they would help to show upon hiring you what skills you were missing in order to succeed.
4) Connect with us. Hit up the people on the websites you've seen. Ask them how they did it. You'd be surprised how many experienced professionals would be willing to help you if you have the humility to ask.
Best of luck and I hope this helps.