Dear @მამუკაჯიბლაძე, I think indeed you may be right and I guess I found the map $\text{hocolim} Y\rightarrow \bar{Y}$ in $\text{Ho}(\mathbf{M})$. The requirement for property (P1), I believe, is exactly that this map is an isomorphism and indeed this should be the right way to interpret that property. Thank you for the suggestion: for some reason, I was so blind to keep up reasoning only at the level of the overlying model categories, without passing to the homotopy categories, but that is probably non-sense.

Dear @მამუკაჯიბლაძე, would you be so kind to elaborate on that by providing a precise answer (or a further comment)?

@GeoffroyHorel Yes, you are right of course, I have just edited the text of the question...So, in the end, do you think that the proposed solution may be the right way (even if not so handy, I would say) to interpret the construction made in the preprint, at least as long as cofibrantly generation is available for our model category $\mathbf{M}$?

Reposting from math.stackexchange.com/questions/1211666/…, as I realized the question may suit MO as well. I hope this is indeed the case.