Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.
But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").
The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.
I don't believe there are that many other examples. But I have never seen any obstruction for this. So:
Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?
Is every presentable $\infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?
Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.