I'm going to sum up what was said in the comments and maybe offer some conjecture, though I don't think it's possible to answer this question with great certainty.
Latin did originally have a verb meaning 'to stand': stāre. Already in the Classical period, this verb also had the meaning 'to stay'.
It's actually not the case that this verb was lost (in this meaning) in all Romance languages: it's preserved in the Eastern Romance languages, e.g. Romanian a sta 'to stay, remain, stand, sit'.
In the Italo-Western Romance, though, the 'stand' meaning was lost. It's probably the case that initially the 'stay' meaning gained primacy, and that's the case still in Italian (stare 'to stay, remain'); in Gallo-Romance and Ibero-Romance languages, it additionally came to mean 'to be', to the point that this is now its primary meaning (French être, Spanish &c. estar; the full paradigms are suppletive, but that's common). The Gallo-Romance/Ibero-Romance shift may have been more dramatic than the Italian one because those areas had more people learning Latin as a second language, or at least in closer contact with other languages.
These developments reflect what seems to be a general move in (SAE?) Indo-European languages away from stative verbs and towards dynamic verbs: verbs are felt as being for expressing carrying out an action, not being in a state. The 'stand' meaning of stāre was stative (it meant 'to be in a standing position', not 'to stand up'), the 'stay' meaning is dynamic ('carrying out the act of staying'); the 'to be' meaning is stative, but the merger with esse still reduces the total number of stative verbs in the language. Proto-Indo-European had a lot of stative verbs, and a lot of these still remain in the daughter languages, but it's not surprising to see their meanings shift to become (more) dynamic (in English a lot of stative verbs are now also inchoative: 'to stand' can also mean 'to enter a standing position'), to see them merge with other stative verbs, or not to see replacements being coined when one is lost.