The TS asks very difficult questions which are not easily answered. It was/is a complicated network of political, social, economical and military reasons and coincidence (!) which changes history.
Rome defeated the Greek sucessor states between 200 and 30 BC mainly because of greater manpower and the typical Roman determination. You can not only say "manipel tactics defeated crappy phalanx", as it is done some times. Roman manipular warfare, copied from the Samnites and other southern Italian enemies of Rome, may be one cause for victory as it is more flexible, but it was not the only one and in my opinion by far not the main reason. Rome could afford defeats and in fact the Romans were defeated many times by their foes (remember Hannibal f.e.) but they stood up again, formed new armies and won the wars due to better economic resources and a huge reservoir of people (and a great moral and believe in Rome).
Why did the Roman empire and the military collapse? That is a question ever disputed since the 18th century. Many reasons occur. Of course I can not explain it but I would like to give some thoughts.
The typical ancient citizen moral declined over the time. Christianity had negative effects because many people cared for otherwordly empires and not the state in which they actually live. In/after the 3rd century AD the Roman empire changed very much. The emperor was no longer the principes but a dominus of absolute power, later supported by christian theory. This led to a decline of identification with the state which became intolerant, dictatorial and oppressive more and more. The regions and local rulers became more important and loyality was more with them and not the empire. The late Roman empire showed in fact many medieval aspects.
Very important in my opinion: there were heavy epidemics in the later 2rd century AD which led to a decline in manpower and degraded both economy and military affairs. At the same time Germanic tribes unified more and more and pressed against the Roman borders. Later steppe people like the Huns added pressure. The empire faced threats which were not known before and had to react on a smaler economical basis. Finally it crumbled simply perhaps because there was no longer the wish to fight for such an arkward construction.
From the 3rd century AD onwards Roman military changed dramatically. The legions declined to local units and the more mobile auxilia became the true fighting force, later resulting in the exercitus comitatenses. Fighting techniques changed according to new threats. Cavalry became far more important. The Roman army was by far the best fighting force in the west still but the borders were long and the menaces abundantly. A great mistake was the later practise to use Germanic or other people not as individuals in the Roman army but as part of foreign troops led by their own nobility and fighting in their own style. Not a good thing for discipline and loyality.
In my opinion there was some mysterious additional cause, less obvious and more on the moral side. Although the individual soldier of 400 AD was probably as good as a warrior (or perhaps even better) as his forefather 300 years earlier, something was wrong. When you look at the battle of Adrianople 378 AD mistakes and low moral among some professional Roman soldiers came into view which could hardly be seen with earlier armies.
HRE: it has not very much to do with the Roman empire in reality. It was the remaining German-Italian-Burgundian eastern and southern part of the Carolingian empire. In state theory though the Roman empire never had disappeared and so the Carolingian state and later HRE claimed to be the Imperium Romanum, similar to the Byzantine empire.
HRE (or better the Imperium Romanum as it was mostly called till the 12th century; Holy Roman Empire = Sacrum Imperium Romanum it was mainly called from the 12th century onwards till the words "of the German nation" were wisely added in the 15th century) was mighty in the 11th and 12th century but even then the denomination was presumptuous.
Roman legions were long gone even at the time of the crash of the western Roman empire and had nothing to do with HRE or the Byzantine empire.
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