To begin with, lawyers tend to see these clauses as essential protections and they are sometimes right. But, right or wrong, they tend to insist upon them, especially in the employment context. This explains their prevalence but, of course, does not necessarily justify their use.
Just to illustrate the cases where they truly are an essential protection, you and a competitor have been fighting for years in court over ugly and untrue things that someone has said about you or your company - non-trivial things that have really hurt you. When it comes time to settle that case, a continuing non-disparagement obligation will be not only helpful but essential to the resolution. The same is true in many other legal fights. When emotions have run high, and parties have antipathy toward one another, it is good practice to help ensure the peace after their fight has been settled to require that they not speak badly of one another and to give a simple mechanism such as binding arbitration to help resolve any follow-on dispute over whether they have done so or not. In such cases, there are excellent reasons to bind parties contractually to restraints on their ability to speak where they would normally be free to do so.
The employment context gets trickier because the antecedent acrimony that characterizes a legal fight may well not be present at the time of a termination and the question then arises: why am I being artificially muzzled? And there is a point to this: why be barred from speaking truthfully about a former situation even if it might be negative? why be at risk of a harassing lawsuit over what it means that something "may" reflect "negatively" about someone? why, in an age of easy communication through social media, be made to feel you cannot even speak about something that may have been a major part of your life, perhaps for many years? What may be seen as a throwaway item by some can be felt to be suffocating by others, and all the more so because it is tacked onto a token severance that gives you very little in exchange.
That said, I would say that the overwhelming number of employers and employees alike see these simply as throwaway items. They figure no one will care about such clauses except the lawyers. And, in most cases, they are probably right. The question then becomes whether one should not sign as a matter of principle or whether to just sign and take the money. Most employees take the money.
Of course, employees can push back if they have leverage. No one is obligated by law to sign a separation agreement. If the terms aren't right, and can be made right, then push back. Insist that the token severance be made more substantial. Or that non-disparagement, if it is to be included at all, be made mutual (it can be quite a head-ache for a large employer to keep control of its many people to ensure that none speak badly of you). Or insist that it be narrowed or clarified so as to reduce or eliminate vagueness about what may or may not be deemed disparaging. Or insist that it be coupled with other considerations that give you benefits apart from your normal final pay, etc. This sort of negotiation can make these clauses a big nuisance from the employer standpoint and may cause the employer simply to drop the clause. However, all of this assumes employee leverage, which doesn't often exist in the routine case, and so, as noted above, most employees simply take the money, accept the restriction, and don't bother to look back.
And so it all depends. For the author of this piece, this was a critical issue. For many others, it is not. Context is critical. And for all but trivial cases, do check with a good lawyer to understand the implications of what you are signing. If the risks are real, there is nothing worse that a harassing lawsuit from a former employer angry with you over some statement you made out of emotion. This is what gives these clauses a bad name and it is also what can make them dangerous. In such cases, be cautious about exposing yourself to such risks in exchange for some token severance. It is probably not worth it.