Yes, it is true that leaving Islam, in an Islamic state, can be a reason for someone to be put to death. Note the caveat: in an Islamic state. We do not have any Islamic state today; only Muslim countries who mix some parts of Islamic law with other types of law.
I think the question you're asking is more about why this might be the case. As you mentioned, Zakir Nike mentioned some points about tarnishing the image of Islam.
Consider also the following cases, as happened historically around the time of the Prophet (peace be upon him): individuals and groups who fought against Islam would enter Islam in the morning, leave it in the evening, and declare it to be false. Not because of any flaw of the deen, but in order to attack and denounce it further. There are some historical accounts of this happening.
One of the benefits and reasons that Allah declared this law, therefore, can be for the protection of the religion itself (which is one of the five aims of Islamic law).
Allah knows best; sometimes we know the reasoning and benefits behind certain laws of Islam, and sometimes we don't. As Muslims, we only need to submit, we do not need to understand the intricacies of each issue and the proofs surrounding it.
In response to some comments about Muslim countries versus Islamic states: Muslim countries are Muslim countries, not Islamic states. Their rulers pick and choose what laws they want -- Islamic, British, secular, or personal whims, often against Islam itself. Will they implement Islamic law, or this particular part of it? I don't know.
Are we all obliged to rule by the law of Allah?
Yes. Allah says:
Translation: But no, by your Lord, they will not [truly] believe until they make you, [O Muhammad], judge concerning that over which they dispute among themselves and then find within themselves no discomfort from what you have judged and submit in [full, willing] submission. (Surah Nisaa, verse 65)
Another verse criticizes the People of the Book for throwing away Allah's law after it was revealed to them:
Translation: [Say], "Then is it other than Allah I should seek as judge while it is He who has revealed to you the Book explained in detail?" And those to whom We [previously] gave the Scripture know that it is sent down from your Lord in truth, so never be among the doubters. (Surah An'aam, verse 114)
Allah describes and depicts the correct attitude of the believers:
Translation: The only statement of the [true] believers when they are called to Allah and His Messenger to judge between them is that they say, "We hear and we obey." And those are the successful. (Surat An-Noor, verse 51)
Islamic law applies to Islamic states. If you're not living in an Islamic state, then you need to worry about local laws, not the laws of an Islamic state.