In a further complaint, does it help you to discredit and dismiss their apology?
No. This will ruin any goodwill that you might have gotten. If you want to remain polite, act like you accept the apology. You gave three examples of when you would like to dismiss the apology: failure to reply promptly, refusing a request for compensation, inconvenience and distress because of mistakes. Let me give you some reasons why you should never dismiss an apology based on these same 3 examples, and explain why you might have gotten the non-apology apology in the first place.
First up: failure to reply promptly. Well, you said you were complaining to a corporation or government. I don't know what 'promptly' means to you, but if I take a look at your other questions, you are at the least expecting a confirmation that your complaint was received, preferably within a day. Let me tell you: that is not going to happen. A corporation or government probably has thousands of messages and complaints arriving each day. If you are lucky, things will be dealt with within a week, but when complaining against a corporation or government, be prepared to wait a month or more before even getting a reply, and be prepared for the reply to be generic, they don't have the time to write everybody a personal letter.
Next: refusal of your request for compensation. In this case, don't dismiss the apology. "I'm sorry, but your request has been denied" after all sounds much nicer then "You're not getting any compensation from us". Read the reply closely. Is there a reason stated for declining? Is there a referral to where you could go and ask for a second opinion? If so, gather all the evidence that you think supports your claim for compensation, like excerpts from terms&conditions, and refute the decision at the appropriate venue. Don't go attacking them based on their apology, instead, base your defense on the reasons your compensation was declined and on the evidence you have supporting your compensation claim. Lawyer up if necessary.
And last but not least: Inconvenience and distress caused by their mistake. Well, this is probably the reason you are getting a generic apology, or maybe even a non-apology. Distress and inconvenience aren't really measurable are they? What may seem like a huge inconvenience to you, might be something that just 'happens' sometimes. Example: A bus breaks down, you miss the train, and have to spent half an hour in a cold, dark night on a platform instead of in a nice warm bed. There's a reason you almost always have to go to court to claim psychological damage: It is a very intangible thing, and you'll need a lot of proof to back up your claim. Sending a generic (maybe non-apology) apology is the right response in this case.
How can you politely convince them that if their apology were sincere, they would have done more?
I'd like to quote a paragraph from the Wikipedia article you linked for non-apology apology:
While the non-apology apology is clearly unsuited to situations where an expression of remorse, contrition, and future change are obviously desirable (e.g. the "happy ending" apology), it may prove extremely useful in situations where little can be done to assuage the apparent offence or prevent its repetition, as when an airline apologises for a delay, in the full knowledge that a future repetition is inevitable. Such tactical apologies may have beneficial effects simply through the validation of the emotions of the offended party: they answer the basic human need for disagreeable emotions to be recognised and acknowledged as important, while protecting the apparently offending party from an expression of remorse. Negotiators often use this tactic to calm tense situations: "an apology can defuse emotions effectively, even when you do not acknowledge personal responsibility for the action or admit an intention to harm. An apology may be one of the least costly and most rewarding investments you can make."
Read it carefully. What does it say? Right, it says it exactly, right there. You get a non-apology if
- They can't promise it will never happen again
- They are willing to recognize your emotions, but really can't remedy anything
- To calm tense situations.
So, they are trying to calm the situation. Recognize that, and thereby their apology, and you will see that you can most effectively work from there.
If your problem isn't solved by the apology, think... Is it really something they could prevent from happening again? If not, don't keep pushing. The other party was right. And denying that is indeed rude.