According to this progression, one cannot become a Vaisnava without first completing one’s eligibility for Karma. Muslims/Christians/Jews are not eligible even to perform the prescribed Karma within the Varna system, because they are outcastes, so how can they become eligible for Bhakti?
Outcaste human beings have every right to practice bhakti. All the scriptures accept this, and Bhagavan Himself has stated it in Srimad Bhagavad-Gita (9.32):
“O Arjuna, women, vaisyas, sudras, and lowborn people who have taken birth in sinful families can attain the supreme destination by taking shelter of Me.”
Here the phrase “taking shelter” refers to bhakti. This is corroborated in the Skanda Purana, kasi-khanda (21.63): Quoted in Hari-bhakti-vilasa (10.106)
“Whether one is a brahmana, ksatriya, vaisya, sudra, or an outcaste, if he has taken shelter of visnu-bhakti, he is considered to be superior to all.”
One may give many quotations from scripture as evidence, but it is important to see what the underlying principle in this consideration is.