when a pillager is summoned holding anything but a crossbow, they will not do anything but stare at you and wander about. however when one with a crossbow is summoned it fires at you. if this is not intentional i would imagine it should run at you and punch you like skeletons do.
this was produced with the pillagers not on my team with the command /summon minecraft:pillager ~ ~ ~ {}. they still do not attack even when attacked themselves, and ignore villagers. I rate this as low priority because this is not likely to happen in the game without commands, however can potentially get in the way of some.
This is intentional. See MC-139468 and MC-139433.