The Bessel functions $J_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights $\ell+1$) of $L^2(\Gamma\backslash H)$. The orthogonal complement of the span of these $J_\ell$'s is continuously (and orthogonally) spanned by the functions $J_{2it}-J_{-2it}$ with $t>0$. This corresponds to the (weight zero and tempered) Maass and Eisenstein spectrum of $L^2(\Gamma\backslash H)$ (of various Laplace eigenvalues $1/4+t^2$). For more details, I recommend Sections 9.3-9.4 in Iwaniec: Introduction to the spectral theory of automorphic forms.

The Bruggeman-Kuznetsov formula is not a weak form of the Selberg trace formula, in fact in many situations it is a more refined (or more suitable) tool than the Selberg trace formula. It can be interpreted as a relative trace formula, see e.g. the paper of Knightly and Li in Acta Arithmetica.

The Bessel functions $J_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights $\ell+1$) of $L^2(\Gamma\backslash H)$. The orthogonal complement of the span of these $J_\ell$'s is continuously (and orthogonally) spanned by the functions $J_{2it}-J_{-2it}$ with $t>0$. This corresponds to the (weight zero and tempered) Maass and Eisenstein spectrum of $L^2(\Gamma\backslash H)$ (of various Laplace eigenvalues $1/4+t^2$). For more details, I recommend Sections 9.3-9.4 in Iwaniec: Introduction to the spectral theory of automorphic forms.

The Bruggeman-Kuznetsov formula is not a weak form of the Selberg trace formula, in fact in many situations it is a more refined (or more suitable) tool than the Selberg trace formula. It can be interpreted as a relative trace formula, see e.g. the paper of Knightly and Li in Acta Arithmetica.