Reverse liaison

As I understand it, many languages are pronounced “incorrectly” in such a way as to make it easier. Liaison in French is very common: saying “les amis” without pronouncing the final “s” of “les” would require a pause or glottal stop. I have just noticed that it is very common in American English to do the reverse in some situations, and I have absolutely no idea why. I have heard many people pronounce phrases like “get off” with a glottal stop instead of the “t” of “get”, and also words like “button” have a break between the two “t”s. This seems very weird – why pronounce words in a nonstandard way when that makes it harder to say, not to mention ugly-sounding? I first noticed it in an episode of Dora the Explorer several years ago, and assumed it was an idiosyncrasy of the voice actor in question. But it seems to be very common, and I even heard a reporter on National Public Radio doing it today.

What theory accounts for this illogical and inefficient behaviour? Wikipedia tells us that it is not confined to the US, which I had known for much longer, having as a child seen too many TV programmes involving Cockney characters.