Brexit has little to do with this particular issue (other than making this bullshit occur when buying from EU companies rather than just non-EU), though I haven't seen anything to disprove it being an insanely terrible idea overall.
It has everything to do with this issue - we'd be following the EU implementation and using their common payment portal if we hadn't bailed out.
Well, it seems that the difference is that using the EU portal isn't mandatory. There is no requirement to use it but the receiver of the good will have to pay VAT + fees if these aren't prepaid. What the EU did by offering a portal is allowing to streamline the process to make it easier for foreign companies to sell goods to EU citizens. And the idea isn't new. Ebay has been doing this much longer through their Global Shipping Program.