So there are two terms floating around here that are easily confused (and are inherently confusing, especially if English is not one's first language). As a native Dutch-speaker with English as a second language, it took me until very late into my BA-studies until I finally had a grip on the difference in meaning.

We'll start with 'modern' historians. Usually - but certainly not always - this refers to historical scholars that study the era of human history that is considered to be 'modernity'. In the taxonomy of historical periods, modernity is understood to be an era that follows up the post-classical (or medieval) period. Generally speaking the emergence of Renaissance thought around 1500 is considered the starting point of early modernity, with 'high' modernity emerging around the 1700s with the Enlightenment, with 'late' modernity emerging out of the large-scale industrialisation and urbanisation of the Industrial Revolution in the early 19th century. Interestingly you mentioned Eric Hobsbawm, who has been a very influential participant in the making and upholding of this taxonomy of modernity.

Now on to 'modernism', or the pursuit of what is considered 'modern'. A key tenet of late modernity was the emergence of ideological thought. Examples of this include most notably the emergence of Marxism (and later socialism), various forms of proto-fascism, anarchism, various forms of social and economic liberalism, and the various reactions to these different ideologies towards the late 19th century. I mentioned socialism and fascism first, because these ideologies - in my view - are most coherently qualifiable as 'modernist'. Socialism and fascism in their European context ironically both heavily focused on the adoption of what they considered 'modern' values regarding social and material culture, and actively opposed political regimes that they considered to be 'un-modern' (e.g. hereditary monarchy in all its forms, a characteristic they shared with anarchism), and actively (and often physically) opposed 'old' structures of capital, labour, and legal rights. With the World Wars the pursuit of modernity took flight, producing horrifying extremes of social engineering, mass displacement, natural destruction, and especially mass murder. Notable examples are of course the Soviet Union, Nazi Germany, Nationalist China (and Mao's China after 1949), Fascist Italy, and to a lesser degree Francoist Spain, Horthy's Hungary, Codreanu's Romania and Poland under Pilsudski. Although these horrors left a lasting image of 'modernity' as a destructive force, the pursuit itself was also instrumental in creating liberal democracies by proxy, furthering the emancipation of women and minorities, and upsetting crucial balances in the realms of labour, law, and politics.

This bring us back to the renowned Eric Hobsbawm. A product of his time, he developed a leftist approach to writing and conducting history during his early career. By the time he had established himself towards the 1950s, he was one of the leading figures in the Marxist branch of modern (!) history, and had clear personal standards with regard to historywriting as storywriting. Most of hist prose is accessible, and it is clear from his choice of topics that he saw history as an emancipatory force for the working class. In this vein he was extremely late to renounce and denounce the Soviet Union for its crimes against humanity, up until the 1980s (I'm not sure) defending the Soviet invasion of Hungary in 1956. In a way you could say he was one of the first major historical 'tankies'. HIs emancipatory use of history makes him a modernist (!), in the sense that he saw society as 'ill' or unfinished in some regards, requiring direct intervention to break up 'old' patterns and relations.

So is modernity over? Short answer; we don't really know for sure. There have been many theorists attempting to create a coherent taxonomy of what constitutes post-modernity, but so far very few of their accounts have survived the upheaval of the first twenty years of this new century.

In all, I hope this helps, even though it is a bit of a lengthy answer. I could post some sources on this later that I used in the completion of my MA degree, if you're interested.