Wearing gloves is all good and well. But what will you tell the boss when the boroscope kit turns to mush and melts due to the heat. Then you WILL HAVE to change the engine. You HAVE to let the engine cool. Not so you might burn your delicate pinkies but so you don't destroy the tooling!!
As an aside, the 757MM states that boroscope can be deferred for (I think) 50 tech log hrs even if there is evidence of core ingestion (smell of cooked chicken in the cabin, blood/feathers on the core IGV's, etc) as long as the engine parameters were normal.
In my experience (20 yrs 757's) I have never pulled an engine due to birdstrike damage, despite having had several go through the core.