I'm applying to PhD admissions to the US. My undergraduate college is in India (non-IIT/NIT/BITS), where professors often ask students to draft their own recommendation letters. How much weight, if any, do these letters carry? Are they even read by admissions committee members?
Of course, the letters will be read and considered. The university ranking as well as how well known the professor is in his field has its weight, but that doesn't mean only the best are considered. The graduate committees typically consider the applications as a whole including the grades, research/work experience, standardized tests, LoRs, SOP, and so on.
Regarding writing your own letter, that's quite common. Professors do as it gives them a great starting point to edit and write a compelling letter without spending too much of their time. It also allows students to highlight different strengths from their letters.