Is it true that the success of a Vedic Homa is dependent on accuracy more than faith?

My understanding is this. If there are three people, one believes that the different devas mentioned in the Vedas are different deities. Another person believes that the different devas mentioned are different names of the same supreme entity. And the third person has no opinion about this topic. The success of the homa will depend on if they follow the process accurately, and not whether they believe in one God or many. The success of the homa is fully dependent on the accuracy of the process, and not faith. Is this correct?