In my experience of car dealers, ‘my mechanic looked at it’ means it hasn’t been done.
Even if it has been done you don’t know when.
My experience of a ‘fully serviced car’ was that it hadn’t been serviced at all, or at least not to fix the things they claimed to have fixed. On the plus side Derby Car Centre ended up paying me the best part of three grand to buy back a car that I bought from them for under two, thank goodness for the small claims process!
When it happened to me though the garage refused to pay me back when I found out their mistake, and it took six months of court process to get my money back and get rid of the car, so in future I would treat anything that a car dealer sa’s that they don’t have a written receipt for as being a lie and so should you. Find another car. Way quicker than fighting them in the courts for months.