I don't have problems secret identities (other than domino mask) I think a small few like Iron Man, Batman because of their Resources and Flash,Superman because of their Power set can work outside having a larger agency cover for them but for most them an Agency or Group who is actively working to cover them makes the most sense in modern setting. To me it make more sense Shield finds out Daredevil and Spider-man are heroes makes contact with them and says we will allow you to be hero and keep your identity safe if you work along with us some times. Then stories are exactly same except that Shield would cover up mistakes they made along the way and make sure there family is safe. Then simple question why can't a hacker get spider-man info or why reporter or over zealous detective finding out secret identity never is able to tell anyone else because Shield suppress it.
It is basically like that in Marvel but they need to more official in how it is done. All the questions of liability, the government allowing this and responsibility are covered by an agency giving heroes some legitimacy in their comic world. Now you might lose the daredevil and spiderman is vigilante angle gain logic about how they are allowed to operate in society who has rules. And simple thing where Shield is responable for tracking every super human activity and when they track them down they give the choice of hero card which allows to be continue being hero or tells them stop makes more sense the current set up.
I think you can get your secret identities stories in a set up like that sort in the same way secret agents stories work. They are plenty of stories where agents have normal life until their cover is blown. You get the normal life and issues that come along with it but you get legitimacy of them being able to hide realistically in the world. To set up better suspension of disbelief you have to tell better lies,The key to good lie is telling some truth in it. I don't heroes being operate themselves is a good lie anymore