I find that it completely unlikely that 'eyeball' planets would have a life-supporting zone.
Over geological time period, the zone is likely to suffer unpredictable and novel variability which is not conducive to developing life.
Over geological time period, the zone is likely to suffer unpredictable and novel variability which is not conducive to developing life.
Earth does that too, though. Ice ages, the odd large impact, etc.
"Life finds a way" - but if the temperature suddenly jumps 100 degrees because your goldilocks zone librated into the sun, then you ain't got much capacity to evolve protection from that.