A cutoff in the linear matter power spectrum at dwarf galaxy scales has been shown to affect the abundance, formation mechanism and age of dwarf haloes and their galaxies at high and low redshift. We use hydrodynamical simulations of galaxy formation within the ETHOS framework in a benchmark model that has such a cutoff, and that has been shown to be an alternative to the cold dark matter (CDM) model that alleviates its dwarf-scale challenges. We show how galaxies in this model form differently to CDM on a halo-by-halo basis, at redshifts $zge6$. We show that ETHOS haloes at the half-mode mass scale form with 50~per~cent less mass than their CDM counterparts due to their later formation times, yet they retain more of their gas reservoir due to the different behaviour of gas and dark matter during the monolithic collapse of the first haloes in models with a galactic-scale cutoff. As a result, galaxies in ETHOS haloes near the cutoff scale grow rapidly between $z=10-6$ and by $z=6$ end up having very similar stellar masses, higher gas fractions and higher star formation rates relative to their CDM counterparts. We highlight these differences by making predictions for how the number of galaxies with old stellar populations is suppressed in ETHOS for both $z=6$ galaxies and for gas-poor Local Group fossil galaxies. Interestingly, we find an age gradient in ETHOS between galaxies that form in high and low density environments.
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