@DavidLoeffler The procedure used can show that there is an injection from the space of $p$-ordinary cusp forms of weight $p+1$ and level $1$ to the space of $p$-ordinary cusp forms of weight $2$ and level $\Gamma_0(p)$. But I'm struggling to show the surjection. I can't show that the $(p+1)$-specialization of the Hida family is an oldform.

@Kimball Using Hida theory, the proof goes like the following: There is a cusp form $\Delta$ (Ramanujan's delta) of level $1$ and weight $12$. So, if we take a $11$-stabilization of it, there is a Hida family passing through it. At weight $2$ specialisation there should a cusp form of level $11$ with trivial nebentypus. But I do not know whether there should be an argument for $p < 11$ or not.

Sorry! I forgot to mention that I'm aware of the result via dimension calculation. My motivation of this question to reach some kind of contradiction using congruences of modular forms.

My comment may not be in the direction of the the "generalisation". I think what's more interesting is the relationship between $H^1$ and the cusp forms. Connection between cusp forms and class group is more Iwasawa theoretic (better to say related to Galois representations). But the previous part is more "geometric". It's what I think about these but I'm no expert.