It wasn’t the means testing that shut it down, if anything it wouldn’t have made it as far as it did without the means testing. As it is, people were already complaining that it was a giveaway to wealthy folks who don’t need it. Take away the income limit and it would have less than zero chance of succeeding.
The president has very little power over legislative decisions. Yes, he could have used the bully pulpit to advocate for abortion legislation, but it would have accomplished nothing.