I haven't exhaustively read the literature on these matters, but I'd say that this is not a settled question.

There are two arguments here: one is that the Roman army essentially "decayed" -- an argument going back to Vegetius in the late 4th century -- with loss of heavy infantry competence, increased reliance on Foederati of dubious loyalty and discipline, centralization of the most effective forces around the emperor's person (against internal more than external threats), leaving the borders to undertrained, underequiped limitanei militias.

The other is that the Roman army "adapted:" the heavy infantry-oriented fighting style was obsolete in a world increasingly dominated by powerful cavalry (see Sir Charles Oman's criticism of Vegetius's assessment of what ailed Rome's military, and description of Adrianople). A mobile, central reserve of comitatenses which could quickly move to troubled regions, while the limitanei fought a temporary delaying action until they arrived, was more feasible than trying to be strong along the whole border at once. Luttwak describes Rome as adapting a defense-in-depth strategy, where the whole of the Empire had fortifications and defenses which could resist raids, rather than a heavily fortified border guarding a defenseless interior, with a breached border allowing invaders to plunder at leisure until the central authorities could respond. In this view, the Romans of 400 AD were stronger than the Romans of 100 AD -- but their enemies had gotten stronger faster than they had. (In something analogous to how the Ottomans of 1900 undoubtedly had an army that was more powerful than the Ottomans of 1500 (bolt-action rifles vs. arquebuses and the like), but other powers had gotten stronger faster.)

... and, I've cited a bunch of sources that serious historians are probably cringing at right now. Vegetius is old enough to be a primary source, of course, but Oman's scholarship is over a hundred years old and much of it is regarded with skepticism. I've also seen a lot of criticism of Luttwak's ideas in The Grand Strategy of the Roman Empire.

I would say that, on the whole, leaving aside some periods of clear strength (e.g., Justinian's reign under the leadership of generals like Belisarius and Narses) and weakness (e.g., the collapse of the West in the 5th century) the Roman armies' changes had a variety of advantages and disadvantages compared to earlier models. I don't think you can clearly say that the late Roman army was a shadow of the Late Republican/Principate legions, but I don't think the case that they had adapted well to new circumstances and performed better than old Republican/Principate armies is strong either. They were different, but I don't think one was clearly better or worse.

There’s a lot of variance between the 4th and 9th centuries. The Roman army underwent major structural changes as a result of the Arab Invasions in the 7th centuries, but prior to that, the army’s structure changed little from the 4th to the mid 7th centuries. This analysis will thus be confined to this timeframe.

In general, modern scholars categorically reject most of the portrayals you’ve stated. This is by no means a slight against you- in many ways, this opinion still abounds in the popular mindset, thanks to a long tradition of negative portrayals of the Later Roman Empire. This portrayal has undergone serious revision in the last few decades.

There are a number of questions implicit in your phrasing which I shall seek to address. One; did the nature of the threat faced by the Empire change during Late Antiquity? Two; how did the army change from the Principate (the period from the first to the mid-third centuries A.D.) to Late Antiquity, and what drove these changes? Third; did these changes negatively impact the performance of the army? Lastly (and perhaps the root of your inquiry); was the Later Roman army as effective as it’s predecessors? Each of these questions could be (and are) the subject of a book, so I’ll try to keep my responses as concise as possible (though I do apologize in advance for the length). I’ll recommend texts for further reading at the end in case any of these topics pique your interest.

Did the nature of the threat faced by the Empire change during Late Antiquity?

There are two main schools of thought here. Broadly, some scholars, such as Peter Heather, believe that Rome faced far greater external pressures during the 4th and 5th centuries than it had during earlier periods. As the barbarians beyond the Rhine and Danube frontiers began coalescing into larger political confederations (such as the Franks, the Goths, and the Alamanni), they presented a far greater military challenge. This increased threat was compounded by the appearance of the Huns, which had the effect of prompting many frontier populations to seek refuge inside the Empire. The Romans proved unable to manage these migrations, leading to these groups establishing themselves inside Roman territory, helping to facilitate the collapse of the tax base (which underwrote the army), leaving the Empire militarily unable to respond to further barbarian incursions.

Other scholars reject this notion, arguing the Empire did not face such overwhelming external pressures. Hugh Elton especially argues that the literary and archaeological evidence demonstrates that there were no significant changes in barbarian society, basic political organization, and military practices. As Elton is a military historian, he emphasizes the continuity of barbarian military practices and capabilities from the Principate to Late Antiquity to argue the Empire did not face increased external threats to it’s security. Adrian Goldsworthy agrees with Elton, and goes on to emphasize how it was the increase in civil war which served to weaken the Empire. These two schools are thus intimately tied into the debate about whether the collapse of the Western Empire was driven mainly by external or internal pressures.

That said, it should be noted the idea of “barbarian hordes” has been largely discredited, even by those who emphasize external pressures. Migrating barbarian groups likely numbered only in the tens of thousands, possessing numbers of warriors in the mere thousands. In regards to the Sassanids, I personally tend to agree with Goldsworthy’s conceptualization of the threat they posed. While the Sassanids were undoubtedly the Empire’s most dangerous foe (evidenced by the large military presence stationed on the Persian frontier, as well as the construction of incredibly expensive fortifications), they were not a peer competitor, at least prior to the 6th century. Persia was not a new “superpower” locked in a deadly struggle against Rome, and was not aggressively seeking a reconquest of the lands of the old Achaemenid Empire.

Scholars are therefore divided on the nature of the threat faced by the Later Empire. While overall threats undoubtedly changed, the exact nature of this change remains debated.