Why would a visitor destroy their travel document?

To expand on Hanky Panky's answer, a country can only remove or deport a person to a country that cannot turn them away, which in reality means a country where they hold nationality and thus have the right to travel documents from.

If the person presents to immigration with a passport and is refused entry, they can be removed by ways of the airline - either a return flight to the origin country, or to the national's home country, at the airline's expense.

If the person presents to immigration with no travel documents at all, they have to be held in the U.K., which increases their chances of speaking to a lawyer, getting legal help, escaping etc.

Also, minors are treated differently than adults - if an 18 year old can claim they are younger, that claim cannot be rejected at face value and the case will be treated as a minor, meaning it's easier to stay. With a travel document, these claims can be rejected easily.

A friend of mine, who was doing part-time work for the French government, was called in to help interview North Korean asylum seekers. Turns out they were not from Korea, North or South: they were Chinese, and didn't speak one word of Korean. My friend, who happens to speak Chinese too, found out they were native speakers of Chinese, most probably from Dongbei.

They had destroyed their Chinese passports after arriving in France, claiming to have destroyed their North Korean passports so they wouldn't be sent back there. Needless to say, their asylum was denied...

Watch UK Border Force, and you will see that many deportation cases turn on whether the person's travel documents can be found. So when they round them up at a workplace, say, they investigate to try to figure out where they live, then they enter that domicile and search it exhaustively.

If they are able to find a genuine passport, it's a straightforward affair -- into the detention van they go, and they are detained and deported fairly quickly.

If they are unable to find travel documents, that puts them in a quandary. They can't deport them to a country they only seem to be from. They must get the person's details, contact the foreign country, and try to get the foreign country to confirm they are a citizen and send over travel documents.

A country like Canada would cooperate, however a country like Bangladesh has a pretty good deal: their citizen is implanted an affluent first world country, making a fantastic wage (by home standards), and sending much of it home to Bangladesh. So there is a perverse incentive for Bangladesh to not help the UK sort out their citizenship.

On the TV program, you often see the Border Force give them a strongly worded admonishment not to seek employment in the UK... And resignedly let them go. Because realistically they do not have the detainee space to hold people for the extended time it might take for the home country to produce.

And the people seek work immediately, of course.

You can imagine the same occurs for people caught at the airport; the government can't detain them potentially forever, so they release them into the general public, with that same stern admonishment.

So for someone illegally in the country who aims to stay, it is definitely in their interest for their proper passport to disappear. I could see a traveler wanting to retain it in a secret place for when they want to travel, but that is impossible at the airport.