  One key ingredient in using galaxy clusters (GCs) as a precision cosmological
probe in large X-ray surveys is to understand selection effects. The dependence
of the X-ray emission on the square of the gas density leads to a predominant
role of cool cores in the detection of GCs. The contribution of cool cores to
the X-ray luminosity does not scale with GC mass and cosmology and therefore
affects the use of X-ray GCs in producing cosmological constraints. One of the
main science goals of the eROSITA mission is to constrain cosmology with a wide
X-ray survey. We propose an eROSITA GC detection scheme that avoids the use of
X-ray GC centers in detection. We calculate theoretical expectations and
characterize the performance of this scheme by simulations. Performing
realistic simulations of point sources (PSs) in survey mode we search for
spatial scales where the extended signal is uncontaminated by the PS flux. We
derive a combination of scales and thresholds, which result in a clean extended
source catalog. We design the output of the GC detection which enables
calibrating the core-excised luminosity using external mass measurements. We
provide a way to incorporate the results of this calibration in the production
of final core-excised luminosity. Similarly to other GC detection pipelines, we
sample the flux - core radius detection space of our method and find many
similarities with the pipeline used in the 400d survey. Both detection methods
require large statistics on compact GCs, in order to reduce the contamination
from PSs. The benefit of our pipeline consists in the sensitivity to the outer
GC shapes, which are characterized by large core sizes with little GC to GC
variation at a fixed total mass. GC detection through cluster outskirts
improves the GC characterization using eROSITA survey data and is expected to
yield well characterized GC catalogs having simple selection functions.
