I disagree emphatically. It would've basically implied that a female hero wouldn't be worth turning to unless her male equivalent were unavailable -- that if the male hero existed at all, he would automatically and inevitably overshadow a female hero. It would perpetuate sexist preconceptions, and it would undermine her as a character if it were suggested that she was only valuable as a substitute for Superman. So it's very important that Superman is, in fact, alive and well and yet Supergirl is still able to be an independent and noteworthy hero in her own right.
And the show has done a good job establishing reasons why they follow their own separate paths most of the time. She was the older cousin on Krypton, the one sent to protect him, so she wouldn't default to seeking his help. They both have their own cities to protect, and she feels she can't be effective and trusted by the people if she can't handle that job without Superman's help. Also, Superman preferentially works alone, and he doesn't work with the government, while Supergirl has DEO ties.
Besides, they didn't have to kill off Steve Austin to give Jaime Sommers her own series. They didn't have to kill off Hercules to give Xena her own series. For that matter, they didn't have to kill off Buffy to give Angel his own series. So why should this be any different?
Sure, audiences are going to ask questions about why Superman isn't around, but audiences always ask questions, and some of them are worthwhile questions and others are just silly and contentious. You can't base your creative decisions on worrying about what bees the audience will get in its bonnet. That would be writing defensively, and as Cat Grant wisely reminded us, it's unhealthy to let your choices in life be governed by fear.