Alexandra actually was often still called Princess Alexandra of Kent, Mrs Angus Ogilvy for a number of years after her marriage
In
the Gazette (the UK's official public record), Alexandra has never been called "of Kent" after marrying. Do you have examples of occasions when she was called that, officially, after marriage? Of course, I understand that the media may have continued calling her by that name despite what Buckingham Palace dictated, just as the media continue to refer to Beatrice and Eugenie as "of York" even though the Palace no longer calls them such.
There was one married princess who was permitted to officially use her territorial designation after marriage, namely Edward VII's daughter Princess Louise of Wales.
The use of The in front of the title Prince or Princess indicates that the person is the child of the Sovereign.
Adding to what Iluvbertie explained, this custom was created by Queen Elizabeth II. Earlier monarchs used "The"/"the" for princes and princesses who were not children of sovereigns; this can be proven by
searching the Gazette.
Mrs. van Vollenhoven is part of the title of Princess Margriet of the Netherlands but she was never stripped of the “of the Netherlands.”
Neither Margriet nor her sisters Beatrix or Christina have ever added their husbands' surnames.

The UK and Denmark are the only kingdoms in Europe where princesses of the blood are forced to drop part of their maiden name/title (or whatever one would describe "of York" as) on marriage and replace it with their husband's surname:
Madeleine of Sweden retained the surname Bernadotte and continues to use it for most purposes; she never adopted her husband's name legally and uses it only for private socializing.
Märtha Louise of Norway has never adopted or used her husbands' names.
Margriet and Christina of the Netherlands retained the [title and] surname "[Princess] of Orange-Nassau" and never adopted their husbands' surnames.
Maria Laura of Belgium retained the [title and] surname "[Archduchess] of Austria-Este (Habsbourg-Lorraine)" (the parentheses form part of her legal surname) and continued to use the unofficial surname "of Belgium" and has not used her husband's surname.
Elena and Cristina of Spain retained the surname "of Borbón" and never adopted or used their husbands' surnames.
She remained 'Her Royal Highness the princess Anne' and also added her husband's name - please don't pretend otherwise by ignoring her title. While I personally am not used to the British tradition of using the husband's first name and find it a bit cringy, if that's the custom in a country, it makes sense for the royals to go along with that tradition.
When living in different countries, I noticed that views on surnames often depend a lot on the tradition you are used to/grew up with as well as views of people around you and decisions they made. Many women in countries where the tradition has always been to either take their husband's name or hyphenate (orders differ depending on the country) are happy to carry their husband's name - versus continuing to (solely) use their father's name.
However, I don't think we will get any further with this discussion as it seems that in your perspective only one solution for all circumstances seems to suffice, so I will bow out at least for now to avoid repeating myself.
It is the British monarchy which insists on "one solution for all circumstances". As you say, most British women seem happy to carry their husbands' names, but the majority of those women probably were free to choose which name to carry when married, and were not forced into the name change.
British princesses may theoretically have a choice according to the law, but in practice, any choice other than giving up their maiden name and using their husband's surname is not acknowledged or respected by the Sovereign or the Palace. Beatrice and Eugenie have been very consistent in
not using their husbands' surnames; since marriage they have referred to themselves as Princess Beatrice, Princess Eugenie, Princess Eugenie of York, Beatrice York and Eugenie York in their professional and private charitable endeavors - without ever adding "Mrs. Edoardo Mapelli Mozzi" or "Mrs. Jack Brooksbank". In spite of the choices they have made when given a choice, all communications issued by the late Queen, the King, and Buckingham Palace post-marriage refer to them as Princess Beatrice, Mrs. Edoardo Mapelli Mozzi and Princess Eugenie, Mrs. Jack Brooksbank.
If Princess Charlotte of Wales in the future truly
wants to become Princess Charlotte, Mrs. John Smith, that is fine by me; what I dislike is that (unless reforms are made) the name change will be automatically imposed on her if and when she marries a man, while it will not be imposed on her brother Louis if and when he marries.
I am not convinced that it is still the "custom" in Britain for women to use their husband's first name. While the "Mrs. John Smith" format continues to be employed in, for instance, the Telegraph's birth announcements, I frankly have never heard of, say, the opposition party leader Kemi Badenoch calling herself Mrs. Hamish Badenoch.
She remained 'Her Royal Highness the princess Anne' and also added her husband's name - please don't pretend otherwise by ignoring her title.
I'm sure meeralakshmi and most if not all readers of this thread understand that Anne remained a Princess after marriage.

British title practices seem to be the most understood by royal watchers in the Anglophone world - to the point where those practices are often incorrectly ascribed to non-British royal families. The use of abbreviations when writing about well-known concepts (with the expectation that most readers will be familiar with them) is not the same as pretending or ignoring them. Nonetheless, as you reminded me earlier in this thread, I acknowledge it is best to write things out in full so that there can be no misunderstanding.