Even though your EE2 installs (that are
2.11.9) should be PHP
7.1 compatible, that doesn't mean that your add-on's are. I know @JCOGS recommended renaming your
third_party as a way to test, but I don't believe that is the best method for testing a base install with all third party software disabled.
Make sure that you have debug in your
index.php set to 1. Next, in your
config.php file, turn extensions off:
$config['allow_extensions'] = 'n';
Now, I would recommend renaming your
third_party folder, but I would also recommend backing up your
exp_modules table, and then truncating it. I believe EE2 also had an
exp_accesories table as well, do the same there. All of these things combined should set your EE install to actually believe their are no modules, extension, plugins, or accessories installed.
Note that if you are still receiving WSOD, you have an issue with error reporting. Regardless of other issues, unless you find something in the Apache/PHP error logs, you should be receiving an error of some sort.