You certainly live in a different world from the majority of the developers that develop applications for customers that can only afford shared hosting services.
Rich companies like your customers may not want to spend money on fixing their code, but a) they can afford it, b) they can pay developers that have a clue like yourself that know how to do the things with the least of the problems.
Software that uses the old mysql extension is not buggy just because of that like you claim.
I am not the one to use the old mysql extension. Actually I was the first PHP developer to ever have created a truly portable database abstraction package which allows you to switch databases without rewriting any code in your application.
phpclasses.org/metabase
If developers used Metabase or any other portable database abstraction package (which PDO is not BTW), they would not have any problems with dropping the mysql extension.
It was the main PHP distribution that encouraged to use the mysql until the PHP 4 days. Unfortunately they created new backwards incompatible extensions instead of evolving the mysql extension. That way, most developers that had applications relying on the old mysql extension were not encouraged to use the new extension, because the old one just worked.
Now it is a bit too late to encourage anybody to rewrite millions of lines of code to use newer extensions.
Your TV analogy is perfect to demonstrate what I am talking about. The fact is the color TV system was backwards compatible with the black and white system. It was a single system. Black and white TV would still show color transmissions just fine, only in black and white.
All this to say that the design mistake was of those that created mysqli, instead of evolving mysql to add mysqli features in a backwards compatible way.
But it is probably not too late for MySQL extension developers to just do that, unify mysql and mysqli extension to provide also mysql extension compatible functions, instead of expecting the PHP developers to write wrappers as you suggest.
Finally, I do not agree that the mysql extension is insecure. What is insecure is the code written some PHP developers with lack of knowledge about security.