Should I use recaptcha plugin?
No. ??
Here’s why: captchas including recaptcha are not effective, they often are not accessibility friendly and don’t really address security effectively. And as you noticed they sometimes do not work.
If you are looking to enhance your site’s security then there are more productive ways to do that.
https://www.ads-software.com/plugins/search/security/
I don’t use any security plugins. I do use Jetpack for it’s brute force login protection.
https://www.ads-software.com/plugins/jetpack/
That protection uses information about brute force attacks on other web sites. When an attacker is discovered by Jetpack on the Internet then all of the jetpack plugin users get that information, meaning if that attacker pokes at more installations then that will be rejected.