- OP is not hiding the fact that he his Indian. It probably shows in his name, on his CV and in his slight accent during phone interviews. Nevertheless, he has been invited for interviews and has even frequently made it to the final round. If being Indian was such an issue, he would not have made it so far so often.
- even in the final round when he didn't get the job, according to OP, he was sometimes offered another role instead. If the HR department was so very racist and didn't like Indians, then why would they do that?
Of course this is not a proof that there was no discrimination at all, but I would say the chances are greater that there is some other issue that is affecting his chances. Apparently an issue that becomes apparent only in the final rounds.
I would definitely not change his name to an americanized version! The problem is that there is something inconsistent between the impression given in his CV and phone interview and what comes out during the final round. Hiding the fact that you are Indian on your CV could only make this problem worse.