The use of homo as a genuine indefinite pronoun is definitely not good Classical Latin, but homo did have indefinite meaning from the very beginning, and by the Christian era Vulgar Latin was clearly well on its way to developing it into a true pronoun (cf. the Vulgate's non in solo pane vivit homo, often translated as "one does not live on bread alone"). Clear lines can be hard to draw, however; it's difficult to find a use of homo that can only be pronominal.
It's probably not the case that homo became a true pronoun before the separation of the Romance languages, but it was certainly close enough already that it only needed a gentle push to become one later on.
Probably the most famous surviving reflex is French on (from the nominative; compare that to the noun homme, from the accusative); for a while it was commonly believed to be the only one, but actually Catalan has hom (with or without the indefinite article un), and apparently a few Italian dialects still use uomo. The heyday of Italian pronominal uomo was the 13th century to the 15th, with only one attestation prior to that, in the 12th. Old Spanish also used omne/hombre (formally unambiguously from the accusative) as an indefinite pronoun, but stopped in the 16th century; it also doesn't seem to have done it before the 12th. The Eastern Romance does not preserve it, if it ever had it.
(Because of the prominence of French on, Germanic influence was often cited as the origin of the pronominal use of homo (cf. Dutch men, German man, Middle English man/men, all 3sg indefinite pronouns originating in the unstressed version of a word meaning 'man'), but given its original indefinite meaning, it's really not necessary.)
Classical Latin typically uses passive constructions (like the gerundives of the other answer) or 3rd person plurals without expressed pronouns (e.g. dicent often becomes "on dit" in French and "men zegt" in Dutch, though English does also render it as "they say") where other languages might use their impersonal or indefinite pronouns.