Scotlands People is the only source for this info. Some points to note. Not all births were registered for all sorts of reasons at that time. Even when they were, many registers have been lost or destroyed over the years unfortunately. The majority of the parish register entries are for the established Church of Scotland, ie Presbyterian. If the family belonged to another denomation, their entries may not be on SP.
Sometimes, whilst you cannot see the birth entry you need, there may be some showing for siblings. Worth checking to see whether any names could help you. There are 24 births between 1795-1810 in Ayr. A number of couples having children then, including a Gilbert and his wife Elizabeth. Note Peter had some children with those given names. However, this couple married in 1803.
Worth noting that sometimes extended family were also having children at the same times (uncles and aunts) with possible clues on names etc. Going sideways can help sometimes when you are stuck.
Monica