(WITH closest_candidates AS (
      SELECT
        ent.id,
        ent.name,
        ent.geom
      FROM
        geo_data.etablissements ent
      WHERE ent.activite_principale = '1071C'
      ORDER BY
        ent.geom <->
        'SRID=4326;POINT (5.4153978921979125 43.271437384501965)'::geometry
      LIMIT 10
    )
    SELECT id
    FROM closest_candidates
    ORDER BY
      ST_Distance(
        geom,
        'SRID=4326;POINT (5.4153978921979125 43.271437384501965)'::geometry
        )
    LIMIT 1)
    UNION ALL
    (WITH closest_candidates AS (
      SELECT
        ent.id,
        ent.name,
        ent.geom
      FROM
        geo_data.etablissements ent
      WHERE ent.activite_principale = '4711D'
      ORDER BY
        ent.geom <->
        'SRID=4326;POINT (5.4153978921979125 43.271437384501965)'::geometry
      LIMIT 10
    )
    SELECT id
    FROM closest_candidates
    ORDER BY
      ST_Distance(
        geom,
        'SRID=4326;POINT (5.4153978921979125 43.271437384501965)'::geometry
        )
    LIMIT 1)

--   UNION ALL
-- [...]  And so on...

SELECT DISTINCT ON (a.activite_principale)
       a.activite_principale,
       a.id
FROM geo_data.etablissements AS a
WHERE a.activite_principale IN ('1071C', '4711D', <3rd>, <4th>, <5th>)
ORDER BY a.activite_principale,
         a.geom <-> 'SRID=32632;POINT(363982.8087 5623158.5124)'::geometry