Actually, the main voting motivation was exactly that, just framed as removing the rights of Europeans to live and work in the UK. Maybe some people didn't know it cuts both ways but that was the prime promise of the Brexiteers.

I recall reading that it appealed to Brits with immigrant backgrounds because they felt like Europeans are having it too easy or something like that.

The other argument was about having a point system instead of letting anyone in, so the European engineers and Indian engineers for example would have the same rights and this will yield in more meritocratic system.

Somehow the companies are not good enough in picking the employees with the correct skillset, for example a British company is hiring unskilled Hungarians as engineers so to fix this the British companies first need to get a permission from the government which will evaluate the person, pay fees and wait a few month for processing to do the right kind of hires.