Admin and site don't display with Blueprint

Hey François! How are you doing?

First I should note some things for you, as recommendations… I don’t know about your server configurations but you’re most likely reproducing them in Local, so here it goes…

  1. PHP 7.0 met its EOL (End of life) Jan 2019, so it’s not recommended to use it, using PHP 7.1 is a good option for compatibility (with hosts) and security updates.
    About it: http://php.net/supported-versions.php
  2. This is more “personal” but the same goes for MySQL 5.5, it is old, somewhat insecure and slow, compared to 5.6 and 5.7 versions. I would recommend using 5.6 and preparing to migrate to 5.7 (I’m doing this right now)
    About it: https://forums.mysql.com/read.php?20,660008,660016#msg-660016

Now, to the issue at hand, can you upload your log file from Local? If you need help with it: How do I retrieve Local's log file?
Also, do you have the WP_DEBUG option enabled? If not, you can add the code below (locally, not live) at the end of the wp-config.php file:

define( 'WP_DEBUG', true );

This will make Wordpress display any errors it catches, so you can have a better idea of what’s happening.
One more question, what page are you accessing when you see this? Are you able to access /wp-admin and login?