Presumably you’re referring to the FOSSGIS OSRM instance’s refusal to route over the A166 approach to Stamford Bridge. OSRM’s default foot profile avoids highway=trunk by default, and the FOSSGIS foot profile does likewise:
My main point is that an access tag matching the default – foot=no in this case – wouldn’t affect router behavior at all. The main consequence is a lack of clarity to other mappers about the intention of that redundant tag.
OSRM avoids the A166 because it’s tagged as highway=trunk and this profile omits trunk roads. I’m pretty sure the omission of trunk from the profile was an oversight during an old refactoring. It happens to work just fine in countries like Germany (?) where highway=trunk is likely to be a high-speed road unsuitable for pedestrians, as well as countries like the US where trunk roads going through urban centres are likely to have separately mapped footway=sidewalk ways, but clearly it doesn’t work for the UK’s urban centres.
For other surface streets, the foot profiles assume access=yes by default.