Step one of gin production is to make a pure alcohol. For example, you could distil from anything that had been fermented (e.g. grape, grain, potato) but unlike say, whiskey or rum, you'd only want the ethanol to come through, cutting any fractions that contained impurities (excluding a little water).
Once you'd got that, you might distil it again to improve purity of the alcohol, before then either diluting with water and soaking your selected botanicals in the liquid, or putting them in the vapour path, and re-distilling for a possible third time, this time not fractionating for purity, but trying to capture the desired flavours and smells that the botanicals provide.
Some gin manufacturers will process one botanical at a time, then blend the resultant "botanical alcohols" to achieve their preferred flavour (also useful for recipe development) whilst others will have an established recipe, but which is monitored since the ingredients are variable based on growing conditions etc.
You could make a gin by soaking your botanicals in vodka (pick your proof), but it won't be stable over time, even if you filter it. The flavour will continue to develop, not always in a good way. Even with properly made gin, the better manufacturers will often try to keep it in final storage for 5 or 6 weeks in order for the final chemical reactions between the botanicals to settle down before selling to the customer, as some people will notice the change in flavour between say, a one week and a six week old bottle of the same product.
Whether the addition of the botanicals is carried out at the same plant as the alcohol production is largely irrelevant, as is the actual source of the alcohol fermentation, except for marketing purposes!