The Romans used archers from the Mid Republic onwards in increasing numbers. Archery had not been a particularly prominent aspect of Western Mediterranean warfare to this point, they instead utilised a great deal of javelin-throwing, a habit that the Roman legionaries themselves never let go of. With the Second Punic War, the Romans began incorporating auxiliaries and mercenaries from other societies that had special skills they did not into their armies. With the conquest of Greece and the East in the 2nd Century BC, this included Cretan archers (Caes. BG 2.7) and probably archers from Rome's various client kings, though these latter must have been regarded as particularly unreliable (Cagniart 88). As the Romans began to campaign more in the East, where archery was a particularly prominent part of warfare and had to be countered effectively if they legions were to easily win battles in the open, the Romans began to utilise archers in great numbers, mainly Syrian and other Eastern auxiliaries (Keppie 185), and even began incorporating cataphracts and horse archers (Kennedy 87-8; Lendon 287-8). Horse archers were incorporating as early as the 1st century AD (James 120). In the Principate, foot-archers were an essential component of Roman infantry tactics, fighting from behind the legions (Culham 240).

I think it is probably true that legionaries were considered as more 'Roman' and more 'manly' than archers, though I don't have any direct evidence for this, and attitudes would have changed over time as the army changed in structure and composition. As a rule, archers don't seem to get a lot of mention in Republican and early Imperial sources. Caesar only really mentions them a couple of times in the entire Bellum Gallicum, even though the Gauls apparently used them prominently (Caes. BG 7.31) and Caesar used them throughout the entire war (2.7, 8.40). As the Romans considered the heavy infantry the core of the army and also the main representatives of Roman manhood through their display of virtus in close combat, they no doubt thought less of archers than they did of heavy infantry. They certainly appear to have found them less interesting to read and write about. However, especially later in the Empire as auxiliary archers and horse archers became prominent parts of the army all over the Roman world, at the very least they were highly trained and well paid professionals that (perhaps more importantly in the eyes of other soldiers) formed an important tactical element of the army.

I don't think there would have been a comparable class divide between archers (professional mercenaries and auxiliaries) and the legionaries (mostly poor men, not always Italian esp. in the empire) as there would have been between peasant archers and more upper-class knights and men-at-arms in the Medieval period. The heavy infantry that the Roman army was constructed around was not really of a 'higher' class than auxiliaries and legionaries themselves featured a lot of racial variety in the Empire. There would no doubt have been competition and discord between legionaries and archers just as there were between legionaries and, say, Batavian auxiliaries, especially considering that auxiliary units were based on ethnic groups, but this I think would have to be a normal thing in such a melting-pot of an army.

Basically, the Romans didn't traditionally use archers because it wasn't a traditional component of the sort of warfare in the Western Mediterranean. But as soon as they got access to skilled archers and came up against enemies that used a lot of archers, Rome began using archers too, and lots of them. They were pretty much always mercenaries (esp. from Crete) and later auxiliaries from the East, as the Romans no doubt saw little point in spending a long time training Italians to use the bow when they could hire significant numbers of experts at any time. As I understand it, this remained the Roman 'policy' towards archers for the rest of the empire. But the Romans certainly saw the use of archers, and incorporated them thoroughly into the army over time.

Sources:

Cagniart, Pierre. A Companion to the Roman Army, ed. Paul Erdkamp, Oxford: Blackwell Publishing Ltd., 2007, 80-97.

Culham, Phyllis. Imperial Rome at War, from The Oxford Handbook of Warfare in the Classical World, Eds. Brian Campbell and Lawrence A. Tritle, Oxford University Press, New York, 2013, 236-260.

James, Simon. The Archaeology of War, from The Oxford Handbook of Warfare in the Classical World, Eds. Brian Campbell and Lawrence A. Tritle, Oxford University Press, New York, 2013, 91-127.

Kennedy, David, "Parthia and Rome: eastern perspectives", from The Roman Army in the East, ed. David L. Kennedy, Journal of Roman Archaeology, Supplementary Series 18, Curshing-Malloy Inc., Ann Arbor, 1996, 67-90.

Keppie, Lawrence. The Making of the Roman Army; From Republic to Empire, Norman: University of Oklahoma Press, 1998 (1984).

J.E. Lendon Soldiers and Ghosts, A History of Battle in Classical Antiquity, New Haven, Yale University Press, 2005.

In response to your mention of A Distant Mirror, I would like to provide a note of caution, if it's not too presumptious of me. While very fine reading by all accounts, Tuchman was not a medievalist by training or practice, and the book is very nearly forty years old, which puts it about two generations away from the current scholarship. I advise you to take its claims with a grain of salt and perhaps consult some specialist literature. If you're interested in the period, I can make some recommendations!

In answer to your (unasked) question, yes, Hundred Years War-era English archers generally came from lower social classes than men-at-arms. But this was not always the case; younger sons of aristocratic families often fought as archers, and successful archers and rich commoners could be accepted as men-at-arms - provided they could afford the equipment. In any case, archers were paid less, at least in part because it's much cheaper to equip one's self as an archer than as an armored horseman, and during the 14th century, men mostly had to equip themselves, unless they belonged to a nobleman's retinue, in which case he was responsible for their equipage.

To what degree did Roman armiess use slingers instead of archers as auxiliary ranged troops? Did the ratios of archers, slingers, and javelin throwers change over time?