There is not a general need for the public to own or use those. Usually people who are at danger have legal ways to get one.
Bulletproof vests are heavy and cumbersone, even people who legally have access to them only uses them when there is a reason to do so. In most of the USA bulletproof vests are legal yet -despite some weird claims- most of the people do not use them.
OTOH, criminals1 have the advantage that they often know beforehand when they are going to commit a crime and if they anticipate the need for bullet protection.
The combination of guns with bulletproofs vests causes considerable trouble to law enforcement agencies. They not only allow criminals to stand for more time during a shootout, but may give criminals the idea that they might survive such a shootout and incite them to refuse to surrender.
While it may seem reasonable to make a vest legal only if the owner also carries a weapon, this kind of regulation does not provide any benefit2. Someone who got a vest with the intent of using it in a criminal enterprise would just avoid wearing it before he is ready to commit the crime. There are no police checks in every corner in Europe to detect when Joe is putting in his vest and also is carrying a gun; the first warning sign would be shots being fired.
1Note that I do not mean one of those apocalyptic scenarios (so common in some narratives coming from the USA) of millions and millions of criminals/terrorists/inept cops wanting to shoot innocent people at each street corner. Most criminals try to get their objectives while avoiding confrontation, and even those who assault their victims have a high chance of surrendering before getting in a full shootout with police force.
I mean the more common and realistic scenario where the criminals that would consider using bulletproof vests because they are ready for a shootout are a tiny minority. But, few as they are, those are the only public who would effectively benefit from the availability of the vests (because, as stated above, they are the ones who know when they will need the vests).
2I found this link explaining that in NY using a bulletproof vest for committing a crime is a crime, which I found utterly amusing. I cannot help but think about some people who happens to be ready to commit a crime and prepares for the possibility of a shootout (possibly with the police), but is deterred from using a bulletproof vest because that would be a crime.