When "Block Sending Corporate Email Via Other Email Accounts" is turned on. It does: Prevent you from replying or forwarding email that was received to you firstname.lastname@example.org account but changing the address to your gmail account. This is particularly helpful if you use the single inbox for all email and in some other situations because your default email address can accidently populate even though that's not the address that it was sent to. This feature will not prevent you from sending email to email@example.com or firstname.lastname@example.org or any other business addresses from your personal email. It just prevents you from doing so mistakenly.
The "Block Third Party Apps from Using Corporate Email", as it states should simply prevent any third party apps from sending email via your corporate email account. Can you download the speedtest.net app on your phone, run the test and then attempt to send your results using corp email using the built in email results function on the app.
Can you confirm that this is not the way that it is functioning?