It’s possible to use google smtp for sending email from the server

Having a properly working DKIM and SPF are not necessarily guarantees that some site administrator will not blacklist you. There are many other lists and processes especially some of the large sites put in play to prevent unwanted user email.

I'd suggest that you contact Gmail and Hotmail and find out why you are being blacklisted and then correct the problem. That is the only way that a blacklist will be lifted .... i.e. by action of the site administrator.

Go to http://postmaster.google.com/ and sign up there, this is recommended by Google https://support.google.com/mail/answer/188131 for blocked emails.

Google provides you there with your individual scoring of your IP and domain.

Verifying your domain also has the side-effect, that you get a higher scoring on your emails.

Also make sure not to send very short (one word) test emails or random strings (like the often used "uname -a | mail ..."), as those are being scored badly.