I would argue that the benefits or difficulties of avoiding taxation are dependent on the rule of law in the states where business is actually done. If the USA or EU took a political decision not to permit such structures then they would disappear.
The USA and EU don't because the benefits to their political class have so far outweighed the problems caused by a lack of revenue, and these structures have developed and proliferated rather rapidly. Additionally the political structures required to create collaborative responses have been developed in response to different issues (the necessity of supporting development in Eastern Europe after the collapse of the wall), and the alignments necessary to operate a larger system in Europe are not yet in place.
I do believe that Brexit will facilitate this, but I think that the process will have to go further to include Macron's ideas on a single budget - a greatly expanded take and disperse for the EU. This will require a huge reform of the Commission. I invite everyone to try and engage with the commission in some process or other and then judge what kind of capability you are engaging with.
I hope that the EU will devolve the distribution of central budget items to locally accountable forums, possibly at the sub national level - perhaps six for a country like France, one for somewhere like Greece. Currently I fear that the favour networks that operate in the EU institutions would greatly distort and corrupt any large fiscal rebalancing.
The big issue is that it seems that several of the national level structures in the EU would creak in the face of this kind of new set-up. Spain and Belgium in particular, but even Italy could be pulled apart by the creation of subnational entities with revenue lines independent of the national entity. Also, what would the mechanism of accountability be ? I think that this should vary according to the tradition of the region, but what are the acceptable limits or bounds on this?
Without Brexit this is simply a non-starter, I think that the UK would have been much more compromised than any other EU state by the emergence of this kind of structure, and I think that the English part of the UK would have been a real problem in terms of subnational accountability - the sovereignty of the commons would have gone and I don't see that as politically possible in England (note the other nations are less touchy about that for some reason!)