Search for shops on major streets

The main problem is indeed to define what is a major road and what is the POI you are looking for. If you can define that in terms of an OSM tag then you can do a simple self-join on placex using distance between geometries and filtering by class and type.

For example, lets say you want all shops close to a highway=primary in Great Britain, you'd do something like:

select shop.place_id, road.name->'name'
  from placex as road, placex as shop
 where road.class = 'highway' and road.type = 'primary' and road.calculated_country_code = 'gb'
   and shop.class = 'shop'
   and ST_DWithin(road.geometry, shop.geometry, 0.01);

If you want to search for all major roads as defined in Nominatim's details view you can also search by rank:

select shop.place_id, road.name->'name'
  from placex as road, placex as shop
 where road.class = 'highway' and road.rank_search = 26 and road.calculated_country_code = 'gb'
   and shop.class = 'shop'
   and ST_DWithin(road.geometry, shop.geometry, 0.01);

This will even be faster because there is an index over rank_search. (The highway constraint is still necessary so you don't end up with things like airports which are also in this rank.)