It would have made that job much easier if there was a tool or technique to pull that cocking piece back into the bolt shroud - once the bolt was out of the receiver - to unscrew from the bolt body - does anyone know what such a tool might look like - or how to "easily" do that, in order to screw that thing back on? I would guesstimate that the Rem 581 is a miniaturized version of the Rem 788 - not sure if any parts are interchangeable - but system at rear end seems very similar.
I am not sure that removing the trigger was absolutely necessary - but it let us look into the sear hole in the receiver and see what was going on in there. Once that cocking piece / striker assembly was removed, the "bare" bolt body appeared to cycle normally in the receiver; then we installed the bolt shroud and it still seemed to function fine - so that left that "cocking piece" / striker assembly contact surface - at that point we discovered the roughness and total lack of lube on the cocking cam of the bolt body. Once polished and lubed, we could not reproduce the "bolt lock-up" - we even fed that fired cartridge into the chamber and "pulled the trigger" - got a second firing pin strike on the casing - and the bolt seemed to open and cycle fine.
I have no clue if that cocking cam on the rear of the bolt body was hardened at factory - as if it might have "galled" - visibly, and by "feel" was very rough - a few passes with 320 emory wrapped around a small diameter rod and then small dab of lithium grease made it feel much different. To be seen if that "fix" works, and how long that will last.