The reason for the executioner being brought over from Calais has everything to do with the differing methods of execution in Britain vs the European Continent. There was a range of methods in each place, but only in Europe were people executed by sword. And only the nobility were usually considered having enough honour to deserve this more honourable method of execution. It's quite complex, so I'll break it down.

Each method (hanging, beheading, burning, etc) had a specific degree of dishonour attached to it. The longer a method took, the more dishonourable it was considered to be. Moreover, if the 'weapon of death' was the natural elements (ie drowning was caused by water, while hanging was essentially asphyxiation, so caused by air), the more dishonourable it was. But at the other end of the scale, beheading by sword which is a) quick and b) done by something human-made was the least dishonourable method by which to die. Plus swords have all kinds of chivalric connotations too.

But importantly, it was your social rank that was the most likely decider of your method of execution. No ordinary person could expect to be beheaded because ordinary people don't carry a lot of honour (so it was thought). The honour of the method chosen for your death was based upon the honour you already carried in life. So nobility got beheaded while regular folk got hanged (or something else dishonourable depending on the crime they'd committed).

But even if you were a noble person in England, swords were only used in Europe, and so you would get the axe. Even Charles I was beheaded with an axe. So Henry VIII's choice to order an executioner from Calais was so that Anne Boleyn could be beheaded with a sword. Although I've never seen his explicit reasoning for this, it could only be (in my opinion) because he wanted to recognise that she was a queen who had been raised at the French court. He wanted to give her the honour that was her due. I'm quite touched by it, to be honest. The fact that he'd already ordered the swordsman over from France before she'd been convicted, well, that's not so great I guess :)

I talk about this and about the meanings of shame and dishonour in my book, Singing the News of Death: Execution Ballads in Europe 1500-1900 (Oxford University Press, 2022): https://www.unamcilvenna.com/author I even talk a little bit about Anne Boleyn's execution ballad in there!