Apology can go a long way, but in my experience (USA) all the successful ones that I have seen were done from a position of strength. That is, the person apologizing felt that after apology his opinions would still be respected and listened to, not dismissed as "another stupidity from that fool". In this case, it could be very powerful. It is also seen as voluntary.
On the flip side, I have never, ever seen an impactful apology that was extracted by demands. Such demanded apology is usually a punishment. Call it by what it is -- flogging (that strong inflict on the week). My 2c.
To be fair, in the language section I point out this is explicitly not using the language of apology (for some of the reasons you bring up), but is instead a more neutral admission of mistake.
On the flip side, I have never, ever seen an impactful apology that was extracted by demands. Such demanded apology is usually a punishment. Call it by what it is -- flogging (that strong inflict on the week). My 2c.