I know exactly how the asylum and migration system work. First an asylum seeker isn't an illegal, it's an asylum seeker until it is determined whether he is a refugee which is a legal status, secondly they are not free to roam around, here I'm talking about legal freedom because unless you only consider physical freedom and thinks that people should be imprisoned without a trial, then we may have a new problem with your points. Now it's true that some administrations lack the means to quickly process asylum seekers which leads to some of them deciding to become illegal residents, the last point being key, when they decide to disappear they become illegals. That last point also negates the idea that it is an accepted situation.
And it's important to remember that this doesn't describe how migration and asylum system actually works in Europe, that's the fringe part that doesn't work and is continuously worked on. Claiming otherwise is why the far rights have actually gained ground by sharing misinformation, the misinformation being that this situation is legally accepted and that Europeans are to be blamed. Now people do not easily become refugees and it often takes at least a year between administrative processing, investigations and interviews to get an answer positive or negative. Europe didn't accept to lose control, Europe has heavy procedures for asylum seekers and migrants which leads many to go on the illegal side of things. And since we are not in Minority Report and every living being doesn't have a chip that allows to know what he will do or where he is at all time some people get through the nets.
Also the large majoirty of immigration concerns economic migrants who are granted visas and are legal residents, it's also fair to mention that among economic migrants a minority are illegals that exploits short term visas and disappear at the end of for example student or tourism visas but again these people aren't accepted which is why the EU area deports hundred of thousands of people every year.