Why do I get the failure «The requested package mager2pro/stripe could not be found in any version» on the «Readiness Check» step while upgrading through the web interface?

It is a bug in Magento 2.
You can workaround it by moving the mage2.pro repository declaration from the global Composer’s COMPOSER_HOME/config.json file to the Magento’s root composer.json file.
I have already done it for your test store (http://lexifir.www67-209-115-135.a2hosted.com):

Another solutution is to use the command line (console) interface for the Magento upgrading.