It does appear Omicron is really less severe (let's not use mild in this context). Studies comparing Omicron and Delta do this in the same environment, over the same time frame, so it's irrelevant that Omicron appeared later.
How does Omicron compare with Delta? Here’s what we know about infectiousness, symptoms, severity and vaccine protection:
Is it less severe? Yes, Omicron causes less severe disease than Delta.
Part of this may be due to Omicron being less able to infect lungs as
it does the upper airways.
The risk of hospitalisation and ICU admissions from Omicron are 40-80%
lower than with Delta.
The risk of death is about 60% less with Omicron than with Delta.
Yet despite the reduced severity, this wave of Omicron has been
associated with higher rates of hospitalisations in many countries
because of the sheer numbers of those infected.
The only silver lining has been how the Omicron wave peaked within a
few weeks in numerous countries, with hospitalisation and daily case
numbers quickly coming down.
One of the linked papers, one example, Omicron SARS-CoV-2 Variant of Concern: A Review on its Transmissibility, Immune Evasion, Reinfection, and Severity:
A study conducted in England observed a reduction in the risk of
hospitalization for Omicron infections when compared to Delta
infections during the study period (Dec 1 to Dec 14, 2021). The
magnitude of the reduction varied according to the inclusion criteria
for cases and hospitalization, ranging from 20–25% when any
hospitalization is used as the endpoint to 40–45% when
hospitalizations lasting 1 day or more is used
So, no apples and oranges are being compared here, the same area, same time frame, and a clear difference is being shown.
And that England study has the following estimation regarding the unvaccinated cases you are mostly concerned with:
||Vaccination or reinfection category
||% Reinfections (cases)
||% Reinfections (hosp)
||HR relative to primary Delta infection in unvaccinated
Which shows that for unvaccinated, the Hopsitalization rate is ~24% lower for Omicron that for Delta.