Do people still actually believe in and trot out that tired old line? There are a million reasons. Language, family, connections, because the UK has (had?) a reputation for being a good place with regards to human rights and upholding the law.
If they were "economic migrants" it would be a damn site easier to arrive on a plane after applying for a visa than to risk their life on a dinghy across the Channel, or stowed away in a van, or hiding wrapped around the wheels of a plane hoping you don't lose consciousness or get so cold you can no longer hold on.
If anyone is thinking "but why are they always young men on their own", I refer you to the pub analogy further up, and also to spend time thinking properly about it.
Your country is at war, bombs, people being killed, or disappearing in the middle of the night. It is 5000 miles to a safe country, a country that used to own yours, and where you know they don't whisk people away never to be seen again. Maybe somewhere that your old neighbour escaped to and you know he'll help you set up a new life if you can just get there Do you make the journey as a young man,knowing it'll be marginally safer than for your wife/mother/sister who would be in constant danger of sex trafficking and rape, then send for them once you are safe so that they can travel via plane, or pack up your entire family and try to keep everyone together and safe, while dealing with people smugglers, hiding from armies and terrorists, and possibly not being able to run away as quickly as if you were on your own.
www.infomigrants.net/en/post/37004/why-do-migrants-try-to-come-to-the-united-kingdom
www.unhcr.org/refugee-statistics/
www.bbc.co.uk/news/uk-53699511
www.amnesty.org/en/what-we-do/refugees-asylum-seekers-and-migrants/
Migrants can still be fleeing persecution but just not fit the terms for asylum applications. We also do not have nearly as many refugees arriving in the UK as certain media sources would lead one to believe.