I don't fundamentally disagree with the points you make Paul, but I notice you ignored the issue of arriving with out papers etc and saying "I'm an asylum seeker just let me in", is a different situation to someone whose has (presumably) been allowed in and has overstayed their welcome...so to speak! The problem of proving who you are and whether you are allowed in becomes monumental when evidence is missing and it takes time.
There is a large number of people in the UK, illegally, and there are many different ways this has occurred, the majority as you say staying on after visa's etc have ended; however, the very visible act of people arriving illegally in full view, of press and of course public, risking their lives and the lives of others has to be stopped, any government will have to stop these boats, Tory, Labour, Greens, Lib-dems, whoever. Maybe other governments have other ways to research individual claims when evidence is missing or unobtainable that will prove faster and more efficient, we shall have to see.