I wouldn't trust any stated preference survey. People often state one thing and do another.
Yes, this is a known problem, however it can be accounted for and more often presents itself where there is a cost to a stated preference. For example in a survey asking for sexual preference can be difficult because young people are often known to have been reluctant to respond accurately if asked in the presence of other family members. Similarly dietary surveys amongst religious groups can get inaccurate results due to the social cost of responding accurately.
This is why as noted above professional polling companies get more accurate results as they have strategies in place to deal with these known problems.
That being said, a well designed survey, even if conducted on the street can still get very accurate results, particularly if they collect enough surveys - and around 2000 seems a reasonable amount - information that allows them to compare the outcome against what might be expected nationally.
I'd hope that there's a technical report which details the survey methodology as well as any benchmarking of the collected results.
Surveys are a fairly specialised field, and whilst I've worked with them a bit, it's been a long time since I've had to work with them professionally.