I was taught Classics at a moderately posh school. And the Roman Empire - which was a very effective nationalist war machine, but utterly barbaric ethically - was very consciously echoed by the British Empire.
It's still consciously echoed in the US. Congress has a couple of giant fasces on the wall behind the Speaker. How much more of a clue does anyone need?
This is no longer obvious in modern Classics departments, and modern Europe draws on more recent history. But it's still a very obvious motivator for imperial Anglophone militarism and commerce. Most obviously in the UK, our recent crop of neo-imperialists not only drop Latin into their speeches but seem to think of themselves as classical heroes.
Actually, I would say that the closest thing you have to R.E. today is the European Union, sans the military might, of course. For all the glory of the Roman legions, it is often forgotten that the Roman empire was a common market first and foremost. Without the economic ties, the empire would have been much more fragile from within.
And the EU of today resembles that ancient structure quite a bit. A huge amalgamation of a babel of nations with a shared citizenship, unification of various laws, regulations and standards so that Portuguese butter is not that different from Finnish, a lot of internal trade reaching even the most outlying extremes of the continent, a conscious building of a pan-European road network, even the original treaty establishing the European communities was signed in Rome.
Interestingly, having semi-hostile Asia Minor (Erdogan) and having almost no control over North Africa is an obvious kind of security problem for the EU, which any Roman general would immediately point out. A secure southern border would be the almost impassable Sahara, not the Mediterranean Sea.
Also, it's very obviously not the case that studying classics makes you imperialist. I don't doubt that some right-wingers may like to add classics into their educational program, to create a link to the past, as you say, but that is not inherent in classical studies.
I studied classics (mostly Latin and Greek) at the university level, and not only was the phenomenon to which you refer not obviously part of the teaching, it wasn't there in the first place.
In any case, the problem isn't the substance of studying Latin, or studying history of any kind for that matter. It's about people who already have right-wing views liking to use classical aesthetics.
And it doesn't help that at that time universities and the study of classics was basically the preserve of the upper classes, who naturally had elitist ideological biases.
There was definitely a sense among many that modernity, democracy, multiculturalism, liberalism were tied up with and responsible for the chaos in Europe at the time. And fingers were pointed at communists, socialists, Jews, liberals, secularists, etc, and in that period nationalists and ultimately fascists or other ultranationalist authoritarians were able to take root. And they often did so by appealing to their imagination of the past.
Ridiculous. People will use anything to justify anything. The fact that some fascists pointed to Ancient Rome is almost certainly just a result of the outsized space Rome and Greece occupies in the western cultural landscape.
The challenge most certainly does not “remain”. Modern classics departments are essentially just history, language, literature, archaeology, and anthropology rolled into one, with a specific focus on the ancient Mediterranean.