Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it? Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic objects in the homotopy category (called $H$-objects). In a certain sense, this is a more correct concept. So my question:

Are H-objects interesting in their own right or is it just some random/pathological extension of the 1-truncation of those objects that really play a fundamental role in modern mathematics?

Or like this:

If we initially thought ∞-categorically, would we invent H-objects? Why / in connection with what?.

The answer to my question could be a list of broad interesting contexts in which exactly $H$-objects appear, but not their higher versions. Or some justification that the concept has really lost its independent value and its role has become auxiliary.
 A: This is not an answer, but slightly too long for a comment.
The main thing I wanted to say is that there is no "correct" or "incorrect" formalization, they just serve different purposes.
For instance (grouplike) $H$-spaces are enough for some purposes (such as proving that a space is simple) and because this is a very minimal amount of structure, this can make them very handy. On the other hand, they cannot be delooped, unlike grouplike $A_\infty$-spaces, and they don't have a very workable homotopy theory, unlike (grouplike) $A_\infty$-spaces.
For your specific subquestion, it's not easy to answer in full generality: the trivial Lawvere theory has very nice up-to-homotopy models, and some simple examples might also have a nice model category that models them. Here is one way to possibly prove that something is not the homotopy category of a suitably nice model category: if $M$ is a model category, $ho(M)$ admits weak pullbacks, that is, for every cospan $A\to B \leftarrow C$, there is a cone $P\to A,C$ which satisfies a version of the universal property of a pullback, but without uniqueness. Indeed, the homotopy pullback satisfies this property.
I want to guess that in many (most ?) cases, for a nontrivial Lawvere theory $T$, $Mod_T(\mathrm{Ho})$ does not have weak pullbacks; but I'm not sure how to prove that, as I'm not sure what the correct way to say "many cases" is.
