From [1]: "Nature is very good at blowing up stars. We [theoretical physicists] are not." That's a reference to the fact that we still don't completely understand why core-collapse supernovae explode. We do know that most ($\sim$ 99%) of the radiated energy from a core-collapse supernova is in the form of neutrinos [2], and we do know that this includes significant contributions from all flavors (generations) of neutrinos [2][3]. The core-collapse environment is so extreme that neutrino-neutrino interactions are significant [4]. According to [5], the core-collapse environment is the only known environment in which significant neutrino-neutrino interactions can be observed.
Altogether, this raises the possibility that neutrinos — and the fact they come in multiple flavors — might have a significant influence on core-collapse supernova explosions. So this might provide an interesting answer to your question, insofar as the existence and features of core-collapse supernovae are important for cosmology. The Standard Model of particle physics is not mathematically consistent unless each generation is complete — which means that if multiple generations of neutrinos exist, then multiple generations of the other leptons and quarks are also required.$^\dagger$
As far as I know, the the relevance of multiple neutrino flavors to the supernova explosion mechanism is still an open question. One recent study [6] says this about the role of multiple neutrino generations:
Neutrino oscillations [which require multiple flavors] do impact the dynamics of the simulations [of core-collapse supernovae], but do not cause the explosion to occur.
Further research may tell us how significant (or not) this "impact" really is.
Footnote:
$^\dagger$ The details of the connection between generations and neutrino flavors might turn out to be more subtle, because neutrinos are massless in the original (renormalizable) Standard Model. Explaining the observed neutrino masses requires something beyond that original model, which could conceivably change the simple "one neutrino flavor per generation" picture. This, like the details of how core-collapse supernova work, is still an active area of research.
References:
[1] Slide 22 in "The Neutrino Mechanism of Core-Collapse Supernovae" (https://www.astro.princeton.edu/~burrows/classes/541/NeutrinoMechv2.pdf)
[2] Page 127 in "Introduction to neutrino physics" (https://cds.cern.ch/record/677618/files/p115.pdf)
[3] Page 5 in "Supernova Signatures of Neutrino Mass Ordering" (https://dukespace.lib.duke.edu/dspace/bitstream/handle/10161/15940/sn_mo.pdf)
[4] Page 7 in "Supernova Neutrinos: Theory" (https://arxiv.org/abs/1604.07332)
[5] First page in "Theory and Phenomenology of Supernova Neutrinos" (https://aip.scitation.org/doi/pdf/10.1063/1.4915560)
[6] Last slide in "The Effects of Neutrino Oscillations on Core-Collapse Supernova Explosions" (https://indico.ectstar.eu/event/47/contributions/979/attachments/705/926/SNCrossroads_Stapleford.pdf), dated 2019
