I can tell you that at least for the fourth century Roman Empire, and probably the entire history of the Roman Empire, this was not the case. The evidence points heavily to the main infantry body of the Roman army undertaking the lion's share of the fighting.

Straight off the bat, it is important to note that your title question actually contains two very different types of units. Skirmishers were light troops which would spread out in front of and beside the main army and would attempt to harry the opponents skirmishers and protect the main army from the opponents skirmishers. The auxiliaries were actually a much more diverse body of troops. In the early Empire the army was made up of two roughly equal parts: the legions, and the auxiliaries. The legions were the classic heavy infantry we think of when we picture the Roman army. They were almost entirely heavy infantry, with only a small detachment of cavalry and artillery with every legion. The auxiliaries supplied every other type of soldier, from cavalry to archers and skirmishers. But the greatest part of the auxiliaries were also medium or heavy infantry. So it is important to not generalise light troops, such as skirmishers and archers, in with the auxiliaries in general.

So, on to who did the most and messiest fighting. We have quite a bit of evidence that the infantry was considered more important. The last fourth/early fifth century army comes down to us through the Notitia Dignitatum, a list of all Roman army units and officers. The units of cavalry are listed before the units of infantry, indicating a higher status of prestige. However, it is interesting to see that that commander of the infantry, the magister peditum, is ranked higher than the commander of the cavalry, the magister equitum. This may indicate that as the infantry had a greater role on the battlefield, the commander of the infantry needed to have the greatest authority. The greater prestige of the cavalry may just be a carry-over from the social class of the Equites, or Knights, which had a high (but lower than the Senatorial) financial requirement.

For how battles went down, Vegetius is a good source, if slightly untrustworthy at times. He says Roman commanders should make use of spies, deserters, and scouts to discover the enemy's weaknesses and their plans and use this to find a favourable location to stage the battle. While the Roman's did not make use of large maps of the whole Empire, Vegetius does recommend small local maps of terrain. Once the place and time of battle were chosen, the army would deploy with the infantry in the middle, with the cavalry on the flanks, and the light infantry (skirmishers) in front of the infantry. Ammianus Marcellinus gives us examples of the light infantry starting the opening engagements of a battle, using throwing spears, bows, or slings. This would serve many purposes: driving the enemy light infantry from the field and thus protecting the main infantry, harrying enemy formations and movements, potentially drawing enemy units (especially cavalry) into a rash attack, and somewhat concealing the movements of the main infantry. When fighting against the Persians, light infantry was also to be deployed against their elephants. As the main infantry lines closed in, the skirmishers would retreat either around or through the Roman infantry formations, taking up positions behind the infantry, and would use their missiles over the heads of the main infantry. This meant they could now be used as missile troops, or as reserves to plug gaps in the line, or counter enemy flanking moves (although they were only used as such in extreme emergencies. The Romans would normally have heavy infantry in reserve already). So light infantry had a very important role in the Roman battle, but it was mostly a supporting role, protecting the main infantry and enabling them to successfully close with the enemy.

Another important type of skirmisher was the light cavalry. They played a very similar role to the light infantry, but on the flanks and designed to protect the heavy cavalry. They could also prove remarkably effective against enemy heavy cavalry. Zosimus tells us that the Roman light cavalry, when faced with the Palmyrene heavy cavalry, feigned a rout, enticing the enemy cavalry to pursue them. Weighed down by heavy armour and weapons, and in the Syrian heat, they quickly succumbed to exhaustion, leaving themselves weak to attack. Light cavalry could also 'pin down' enemy infantry by charging at them. The infantry would have to form a tight group, most likely in a shield wall kind of formation, limiting their movement. If morale was low, they might break and run. If not, the light cavalry could wheel to the flanks, hitting them with missiles, and form up for another charge. They could repeat this process until the main infantry could catch up, or the troops broke. Once the enemy broke, light cavalry could be devastating in pursuit, mercilessly cutting fleeing men down. There was a huge disparity between the victor's casualties in a battle, and the defeated's casualties, and this is mostly from men being killed as they fled. While this might have resulted in the cavalry obtaining more kills per battle than any other unit, this would have been 'easier' cutting down of fleeing enemies. The main fighting was still done by the infantry.