Word won't get around. You won't get a job you don't apply for though. And if they just do skills tests without any other consideration they're probably not an ideal workplace.

Everyone wants senior level guys in React and other frameworks I do not have. (Though I am currently teaching myself React and do know a good bit now.)

Of course they do. Everyone wants React guys with 20 years experience in it, Java guys with 120 years experience. And they'd like to pay in aspiration and a twice yearly packet of macadamia nuts. But they can only choose from those who apply with all their varying experience and salary demands.

If you've got 10 years experience you've got enough to apply for a senior role (arguably more than enough in many places) and if you've got your head around the basics of React you can tick that box too. Just be honest that you've looked at it outside of work, understand it but haven't had the chance to use it professionally yet. Senior roles may be preferable to you, where there's less emphasis on churning out code and more on getting the most out of the team as a whole, as well as rewards commensurate with your experience.

I saw a guy interviewing on YouTube the other day for a React job who didn't know the difference between componentDidMount and componentWillMount and the interviewer seemed to think he did well - so if you've got that far, you're good to go!

Just apply for enough roles and you'll find one that works for both sides. Don't be put off by barriers in your own mind, make employers tell you you're not the right fit and ideally why, then move on until you land one.

Good luck.