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I am not sure what your criteria would be for a proof to be given a geometric interpretation, but the reason why weights "disappear" when we take the inverse limit on the level stems from the contraction property of Hecke operators (at $p$), or informally from the fact that Hecke operators at $p$ diminish the level.
As you know, the proof of the isomorphism between the two different Hecke algebras requires the definition of a map between Hecke algebras acting on forms of weight 2 and forms of weight $k$. Because theses two algebras are sub-algebras of endomorphisms generated by the same abstract elements but acting on different objects, this amounts to constructing a map between the cohomology of (one of the level of) the modular tower with coefficients in the constant sheaf and (one of the level of) the modular tower with coefficients in a sheaf of weight $k$ (or the same thing with the modular tower replaced by the Igusa tower, as in Kevin's answer). This last map is really no big deal: if memory serves, on the sheaves it is just projection on the last component. The remarkable fact is that the map on cohomology then is surjective with a finite kernel (and is an isomorphism in the ordinary case); the proof of this assertion being exactly the contraction property. Note that the proof necessarily requires the choice of a level at some point; how else would you even state the result?
Note for instance that for a tower of more general Shimura varieties, it is not at all obvious how the contraction property will play out: group-theoretic properties of $\operatorname{GL}{2}(\mathbb Q{p})$ \operatorname{GL}_{2}(\mathbb Q_{p})$ really do play an important role in the proof. See the reference below though, for an answer to these questions. So in the end, the isomorphism between the two Hecke algebras seems to me to come from the interplay between the cohomology of modular varieties and group-theoretic properties of the Hecke algebra. A very general formulation of this fact can be found in D.Mauger Algèbres de Hecke quasi-ordinaires universelles. Ann. Sci. École Norm. Sup. (4) 37 (2004) (section 2.4 to be precise) 1 I am not sure what your criteria would be for a proof to be given a geometric interpretation, but the reason why weights "disappear" when we take the inverse limit on the level stems from the contraction property of Hecke operators (at$p$), or informally from the fact that Hecke operators at$p$diminish the level. As you know, the proof of the isomorphism between the two different Hecke algebras requires the definition of a map between Hecke algebras acting on forms of weight 2 and forms of weight$k$. Because theses two algebras are sub-algebras of endomorphisms generated by the same abstract elements but acting on different objects, this amounts to constructing a map between the cohomology of (one of the level of) the modular tower with coefficients in the constant sheaf and (one of the level of) the modular tower with coefficients in a sheaf of weight$k$(or the same thing with the modular tower replaced by the Igusa tower, as in Kevin's answer). This last map is really no big deal: if memory serves, on the sheaves it is just projection on the last component. The remarkable fact is that the map on cohomology then is surjective with a finite kernel (and is an isomorphism in the ordinary case); the proof of this assertion being exactly the contraction property. Note that the proof necessarily requires the choice of a level at some point; how else would you even state the result? Note for instance that for a tower of more general Shimura varieties, it is not at all obvious how the contraction property will play out: group-theoretic properties of$\operatorname{GL}{2}(\mathbb Q{p})\$ really do play an important role in the proof. See the reference below though, for an answer to these questions.