@andypandy You are missing the point spectacularly, and I thought I was the one with English as their second language….
The UK government cannot refuse dealing with an asylum application solely for formal reasons, whether the person claiming asylum arrived legally or illegally. They can of course refuse asylum and deport the applicant, but they have to hear their case first. This is an essential part of the 1951 refugee convention referenced in the opening sentence of your linked document.
The point of your document merely is that the UK embassy in, say, Damascus, is not obliged to issue travel documents to someone they suspect will be entering the UK for the purpose of claiming asylum there.
Le Tocquet does the same thing: If the immigrant is stopped by UK border control while still in France they cannot claim asylum in the UK, as they will have to get there first. Once an immigrant reaches Dover the UK government would be bound to hear their case.
Whether they arrive legally or illegally is irrelevant, anyone can claim asylum in any signatory country of the UN declaration of refugees. This is the reason inter state agreements like Dublin or Le Tocquet or extraterritorial transit zones at airports exist.
Nothing at all in your document says that an asylum seeker must seek asylum in the first country they reach, in fact such a constraint would not be permitted under the 1951 convention. In fact, that would not at all be practical. Do you seriously propose that the millions of poor refugees from Syria should be dealt with by the neighboring countries like Jordan, while only the wealthy who can pay for a flight or hire a yacht that takes them to directly to Cyprus or Greece are free to claim asylum?
Nevertheless, states in common travel areas like Schengenland can make an agreement (between states!) that the first state registering the refugee is stuck with them administratively (an amazingly stupid idea that of course had to fail), and that the states collectively will only hear one application (a better idea).