The only Zoroastrian community that does not allow conversion is the Parsi one in India. They are very protective of their community for historical reasons.
The Parsis are descendants of those original Persians who escaped to India after Arab-Islam invasion of Iran centuries ago. India accepted them at the time so long as they didn’t proselytize. Equally, the Parsis wanted to preserve their ethnic and religious identity in a foreign land, so did and still do not welcome converts.
But theologically there is no foundation for not accepting converts to Zoroastrianism. After all, when Zoroaster started preaching, no one was born a Zoroastrian :)
I’m not sure if it was the Avesta community that discouraged conversions but it seemed quite cut and dried. I agree with you on the no original members at the beginning thing!