Consumption taxes aren't regressive unless you take a very narrow view of what happens in given year. Consumption taxes are linear as the money will be spent at some point and taxed there (and if it isn't spent it's fantastic as someone just provided free services to society and didn't want anything in return). Still if you are concerned about them being regressive in small time window then there are many ways to compensate for that:

-tax necessities at lower rate -give every citizen some way to get some of the consumption tax back (up to X amount) -introduce more social programs like some kind of unconditional payment; as this will be much bigger % of income for poorer people it will in effect make consumption tax progressive.

>>The truth is there isn't a 'simple' solution to this, which is part of the reason that status quo is unchanged (another part of the reason is obviously the lobbying power of large/multi-national corporates.

I think your "another part" is what is really going on here. I would add one more reason: people are not good at understanding economics and consequences of regulations. Even if you prove on paper that consumption tax is just more fair, will make companies pay in market they operate and therefore will lower tax burden on lower/middle class it's still easier for a politician to shout: "tax the rich CEO's!" and they will get people to follow them.

Robert H. Frank's Luxury Fever[1] advocates for a progressive consumption tax, and has some fairly compelling arguments. One of which is that it flips the "save money" incentive to consumers. This coupled with Piketty's wealth tax would be difficult, but probably effective in combatting wealth inequality, and making taxes a bit more "fair"

Basic Income combined with a Prebate seem like they would work rather well in consumption-tax land.

Until you start dealing with the fact that the largest taxpayers really don't consume that high a proportion of their annual income, and someone has to pick up the bill...