My own experience is that someone asking for a specific product is a rarity. The skill of the tech. is another matter. If you are good at what you do, your Clients like you and you act professionally the clients will come (In my opinion, well actually in my experience as well).
I have absolutely no idea what brand of colour my hairdresser puts on my hair, or where they were trained. If the end result is good, then I will follow them from Salon to Salon. If it is not, then I change hairdresser until I find one who suits me, and if we are all honest most of us do the same.
Most lone techs do business by word of mouth (ie recommendation). If whatever product they are using suits them and they are happy with it, great - but it is my belief that it is primarily the combination of skill and personality that makes clients come back for more.