Overnight stays in London by nationality 2018:
USA - 3,87 Million
France - 3,69 Million
Germany - 3,26 Million
Ireland - 2,78 Million
Spain - 2,53 Million
Netherlands - 1,95 Million
So, from the top 6 countries of origin for overnight guests in London 5 are EU countries.
According to the reasons for the visits of EU citizens to London:
5,688,000 Holiday
3,210,000 Visiting friends or relatives
2,376,000 Business
561,000 Miscellaneous
At the point of entry you do not know why a person is entering the country - and if you change the entry conditions for one European then you do the same for all. It will not be as easy to visit the UK after brexit - which you know very well. How many people are going to apply for visa free status weeks in advance for the purpose of spending a weekend in London, when there are other possibilities such as Paris or Vienna, which just involve getting on a train ? Or are you going to stamp the passports which most Europeans don't have, on entry ? Britain already has over a million illegal immigrants from non EU countries - maybe you should sort out this problem before adding to the number.