Biometric safes. Unlike a car, adding something directly to the firearm itself will cause a potentially fatal decline in reliability. As for firearms manufacturers themselves, I believe they've made most handguns far more drop-safe than they were 50 years ago. But I'm not expert on the evolution of firearm safety mechanisms.
In your conception the 2nd Amendment has been repealed? Otherwise, that's not how rights work. It's hard to change minds with (seemingly) highly flawed ideas like this one. Don't blame the sensible people who actually think it through. But maybe I'm wrong and you have. In that case can you answer a few questions?
How have you factored in the likely dishonesty from the relatives? Who gets to pick who vouches? The person trying to exercise their rights or the government looking to deny them? I'm trying to imagine a guy wanting a gun, his wife telling Uncle Sam he's too dangerous, yet not so dangerous she doesn't leave him. More realistically, the wife is going to consent. Woman are always claiming how they feel pressured to go along to get along.
Let's say you try to mitigate instances where people vouch for people they consider dangerous by enacting a criminal punishment for anyone who vouched for someone who subsequently used a gun to commit a violent crime. Then people are going to cover their own ass by not vouching. Maybe would rather 10 good men get denied in order to hope to prevent one guy from acquiring a gun for the purposes of misuse? To me that's the opposite of American justice where it's better 10 guilty men go free than an innocent man be jailed.