When I read the passage in LASORS about the "training flight" I find it very easy to take from the paragraph that we as instructors should be assessing the skill level of the person we are flying with, then according to the paragraph we should "interject if necessary to improve on these".
If by the end of the flight, the standard has not got any better, then for me, the aim of the flight hasn't been achieved. After all, it is a training flight not a test, so if the aim of the training flight is to increase skill levels and that hasn't happened, then this section becomes relevant "Where the aim is achieved the FI will sign the applicants logbook, append his/her licence number and identify the ‘Training Flight’ for the examiners purpose."
Why, because it says if the aim is achieved, then you sign, if it isn't, then you don't. I would far prefer it if that was made obvious, but I think the inference is very clear from that, otherwise why mention the phrase "where the aim is achieved"?
In the hundreds of logbooks I have seen, the only time I have noticed each training flight to be signed by an FI is for people who have flown in the States. I simply don't know anyone who signs after every flight, as I mentioned before it is messy and unnecessary. I have taught in schools in the South of England, the Midlands, Welsh Borders and up here in Scotland and not one CFI has insisted or even mentioned that I should sign a students logbook after each and every flight. The only time I've seen lots of signature like that is for P1/S in an airline environment.