Protect access to a whole website page with reCaptcha

I know of the traditional method using .htaccess file to protect my website's access using user name and/or password, but it's quite ugly.

Can I instead protect access to a whole page of my website (containing personnal information) using a reCaptcha validation checkbox instead?

Of course I looked it up on Google, but I keep being pointed to articles teaching how to do it on forms, which is not what I want...

I know some websites do it using something called CloudFlare, but it appears to be pay...

My website is in classic HTML/Javascript/jQuery/PHP.

Thanks.