EU wants to hinder rascal states that make money at the expense of the rest of the union taking advantage of their small size: they won't be able to collect VAT for business made in the whole EU just because they offer a lower rate. (I mention the size because the deal is convenient for the state only if its economy would be otherwise very small).
It does not matter if the company is American or European, I could have made a Luxembourg subsidiary of an Italian company and paid Luxembourg VAT rates.
Of course, this is a very stupid way of doing it: it works only for VAT, while the countries can still be fiscal havens for any other tax, and moreover it makes life hell for small B2C business.
The proper way would just agree on a uniform tax policy. Never going to happen.
They should "just" add a size limit or at least make it much simpler for companies to comply with the law.
Unfortunately not all EU countries have a lower threshold and the new arrangements have a lower threshold for mandatory VAT registration of any amount -- a single €0.99 sale in Estonia and whoops, you need to be registered to collect and pay VAT in Estonia or via your own tax authority's One Stop Shop.
If the £81,000 threshold applied to the new arrangement, nobody would be shouting. But as it is, this will kill a huge number of spare bedroom businesses and start-ups.
I registered my small business for VAT voluntarily when I was bringing in much less than 81k since I found that larger companies wouldn't deal with me if I wasn't VAT registered. Since I already had an accountant and used software for my bookkeeping the effort and increase in costs was negligible.
What is difficult is charging a different VAT rate for every non-VAT registered individual in every country you sell to.
Would of course be done through a service. Maybe it's time for these "innovative" payment processor to actually be innovative and provide this service.
"£81,000 threshold applied to the new arrangement"
It can't apply to the new arrangement since you are now paying taxes in the customers country.
It seems that you would just either do your local sales as before and your international sales through a service or start two companies. That why you should be able to keep your threshold for domestic business. (Don't quote me on this though since I'm not in the UK)
Edit: yes, I meant VAT "reductions" not "payments". My point holds though: there won't be any more VAT incentives to move.
So if you have $0 VAT from EU purchases, but $40 VAT from sales, you pay $40. But if you paid $40 VAT from EU purchases, and have $40 VAT from sales, you pay out $0.