Such a simple question, but with a complex answer!
Yes, the list allows you to build Later Italian Condotta armies.
However, in the later period (15th century, as opposed to 14th century) the Condottieri were "Captain Generals", hired as individuals by the state in question, rather than the state hiring one or more Captains and their mercenary bands. The state then put the army together and the Condottieri commanded it. The state had its standing army (guard units and militia), raised Provisionati for the campaign, as well as hired the mercenaries directly. The vast majority of these mercenaries were Elmeti, mounted armoured troops, known as Lanze Spezzate.
This was different from the earlier period (say late 13th century and 14th century) where the Condottieri were "Capitano di Ventura", mercenary captains leading their own mercenary bands who would hire themselves out to various states. Some of the well known Capitano di Ventura include John Hawkwood and his "White Company", Werner von Urslingen and his "Great Company", Lodriso Visconti and his "Company of St George". All of these existed in the mid 14th century and could be quite large at the height of their power - the Company of St George consisted of 4,000 to 6,500 men at its peak.
If you are creating an army based off one of the Capitano di Ventura in the 14th century, then the White Company list is a good start. For one of the German companies, I would use one of the two Holy Roman Empire lists. For a later "German" army, use the Later Holy Roman Empire list to build a Landsknecht army.
Finally, the Later Italian list can also be used to build a list with options. Take an unnamed state as your core army and add the "unusual" units as allies (i.e. you want a unit of Longbowmen, then add them as a unit of Neapolitan allies).
The above is likely (in true wargamer fashion) a gross oversimplification....