The answer to this question requires vast historical and linguistic knowledge. You should definitely read history of Persian, at least
an introduction to it.
After Islam came to Iran,
Perso-Arabic alphabet was developed and Persian language flourished even more then, producing world class poets, scientists and scholars.
We do have an alphabet, in old and middle Persian:
Old Persian inscription:
Middle Persian Alphabet:
But we have been using this writing system for nearly 1200 years and I believe there is no shame in it. Too many historical books have been written in this system for 1200 years and changing the whole alphabet system doesn't worth all the difficulties and obstacles in this way and it won't change things much either.
Look at European languages, all of them have their own great histories, but most of them use the same writing system, Latin, with minor changes and they are not after changing it, because it has assimilated in the society and culture so good that it's nearly impossible to change it in modern times and even if it's possible, it would take many years and huge costs bearing no significant fruit.
Many languages in the world have borrowed from other languages, and became better and better. Persian too, as much as it influenced almost all the languages in western Asia, it also took some of their advantages.
I can't judge if Ataturk's decision to change the Ottoman alphabet in to Latin one was right or not, but I know he faced great difficulties and achieved something spectacular, raising a whole new generation who could barely understand the language used in the empire that existed in their homeland no more than 40 years earlier.
So I can ask you, why don't you use your own alphabet and instead use Latin (ignoring few changes that was applied to Latin to create the Modern Turkish Alphabet)? The answer is that, it just fits well in the society, why change it? No one forgets the great histories of our countries (Iran, Turkey) even if we use another writing system. It doesn't matter much.
PS: The right spelling is Charshanbe (Chaharshanbe) and Pan[j]shanbe (Thursday).
But to ear, they are heard as charshambe and panshambe.