In August 2018, Refugee Rights Europe published a report entitled The Long Wait which surveyed 870 individuals living in the informal refugee camp in Calais, roughly 15% of the camp's population. Of these, 94.6% reported that they were aiming to continue on to the UK in order to seek asylum.
In total, 40% are hoping to reach the United Kingdom as they have
friends and/or family there. 14% cited that they hope to reach the UK
as British asylum laws are preferable to the French laws, with a
number of respondents telling researchers that they fear their
application would be rejected by the French authorities but
potentially accepted by the British. A further 23% of respondents told
researchers that they are looking for refuge in the UK because of
their language skills.
Another large reason, related to language skills, appears to be aspirations for better opportunities in the UK. The survey asked respondents what they thought they would work as in the UK, and the results show a clear preference for skilled work and higher education.
This could be another reason these roles are not sought in France, as the survey also found that English proficiency was far higher than French. Presumably, a contributing factor is that migrants' French is often insufficient to pursue these career paths.
Finally, you have mentioned in your original question and comments that migrants may be put off by societal attitudes towards migrants. However, the survey also found that residents were afraid of 'facists/racists in Calais', as well as a fear of the French government, and especially the French police:
One of the most significant causes for concern was distrust of the
French police, who many believe are “very violent” towards the camp’s
inhabitants. “The French police treat us the same as Syrian police,”
said one resident. Some expressed frustration at police blocking the
exit from the camp at night, reporting that their “freedom has been
In fact, more respondents were afraid of the French government than the British - 44.7% felt 'a lot of
fear' towards the French
government, compared to 12.2%
towards the British.
Admittedly, the fieldwork for this report was conducted in February 2016, before the Brexit referendum. However, the two follow-up reports, Still Waiting and Still Here, which were conducted after the Brexit vote, exhibit similar responses. For example, 73% of adults and 75% of minors thought that the UK was the 'best country in Europe for them'.
In particular, on the issue of Brexit, respondents were asked in Still Here whether they had heard of the vote. Approximately two-thirds of respondents had. Of these, almost twice as many respondents thought that the vote would affect them in a positive way (29%), rather than a negative way (16%).