First of all, the system as described in the article is already effectively in place: once you get your green card, you have to wait 5 years (3 if it's through marriage) before you can naturalize. In that time - or more, if you don't apply immediately - you pay taxes etc, but you cannot vote.
It's even more amusing with non-citizens who aren't green card holders, because not only they still pay all the taxes, but they're denied most of the benefits those taxes fund (so e.g. you still pay social security on your wages, but you don't get to actually claim any payments when a citizen could).
So I don't think there's any insurmountable political obstacle here.
But even beyond that, "taxation without representation", as originally used, didn't actually mean voting rights per se. The complaint, rather, was the lack of anyone specifically representing the interests of the colonies, because they were basically arbitrarily assigned to districts in Britain proper for the purposes of parliamentary election. So not only colonials didn't vote, but their MP would typically never even set foot on the territory he supposedly represented... which is why it was pointed out that it's not really representation.
However, representation was not equated to vote - keep in mind that the original franchise wasn't even universal among white males. However, those that couldn't vote were still deemed to be represented, on the basis that they lived in the same district as the voters.
Even today this principle still applies: while only citizens vote, the number of congressional seats, electors etc is calculated on the basis of the entire state population as of the last census, which doesn't distinguish citizens and non-citizens. So areas with large non-naturalized immigrant populations effectively award more voting powers to their resident citizens to "represent" the rest of the district. And this practice was explicitly blessed as valid by the Supreme Court in Evenwel v. Abbott.