It would be extremely unlikely for a Rabbi to do that for specific theological reasons that I'm not going to get into at the moment. (Essentially even most Orthodox Jews are functionally close to universalists in regards to the afterlife and don't think that non-Jews need to convert to go to heaven.)

There's also a clear distinction when one is talking to a relative like the OP mentions as opposed to being a stranger talking to essentially random people. As a society we consider that to be much less of a problem.

But besides all that, I don't really have a problem with the Rabbi or Priest who does that. If they sincerely think that the stakes are just that high then they should do whatever they can to get people to convert. Someone who stands by while someone dies and they maybe had an opportunity to do something that they think will save them seems morally reprehensible. The problem with the person trying to get deathbed conversions is more because a) pragmatically it will likely have more of a negative reaction to surrounding individuals and thus hurt one's cause more than it helps b) the religions are simply wrong, and thus taking up the last few precious minutes someone has in this world.