If you have 4 or 5 hp then sure, you're probably feeling afraid for your character. - But if the DM had just rolled a little better on that last hit you took, you wouldn't be feeling afraid for your character, you'd probably be feeling frustrated and annoyed as you start generating a new one.
If you die at 0, at what point do you start feeling afraid for your character? At 30hp a crit or failed save and an above-average roll could kill you before you even hit bloodied.
So you're afraid most of the time? Or are you just apathetic, because you're always at risk?
But when you do get low, what are your tactical decisions? 5e doesn't have many effective defensive options outside of spells. Running will generally rely on the rest of the party to defend you, risking themselves, since many monsters can run down the average PC. Dodging increases your chance to live by a little, but again at the expense of your contribution to the fight. The only real option, if you're playing a character with resources, is to burn more of them, if you weren't already.
The sort of people who care about their characters enough to be your target victims probably wouldn't enter a fight casually, or with no tactics. They're already trying to minimise their damage taken vs effectiveness ratio.
Even with the death saves system, people still fear the results of getting unlucky in a fight: There is no guarantee of survival, and no one enjoys sitting on the sidelines rolling death saves. Its not as bad as having to roll up a new character, but its still not a fun way to spend leisure time intended to be devoted to trying to de-stress.