Saying it in the Adhaan is not wajeb, and it is not a part of Adhan. It is a recommended add-on. Whenever we mention the Prophet (s), we mention Imam Ali (a) as well. This is similar to you saying “jalla jalaluhu” after the Mo’adhen says “Allahu Akbar.” You are glorifying Allah after hearing his name. So it’s not a mandatory add on. It’s a recommended add-on.
For more details you may see this video at minute 11 and onwards
https://youtu.be/en-JoTh9RPQ
1- There is a hadith that says Bilal said it at Ghadeer, but the chain is weak. So some scholars accept this hadith and those who are strict with the chain may not accept it.
2- The Imams didn’t prohibit it. But due to persecution, they and their Shias could not publicly say it. Imagine if today in some Sunni societies you can’t say it, then how it must have been in the past.
3- There are hadiths that recommended the third shahada (not in Adhan, but in general). So scholars take the generality of these hadiths and apply it to the Adhan as well:
A- You are allowed, by Islamic law, to say recommended things in Adhan.
B- The third Shahada is recommended.
Conclusion: you can say it in the Adhan.
Shias can take general recommendations and apply them to various situation to support Imam Ali (a). This is not an amendment. It’s simple an application of a general recommendation.