An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism.  For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^*$-oriented bundles of the sort wanted, modulo precision about condition (ii). [Senior moment nonsense eliminated].  You are studying the J-map $BO\to BF$ (or $BU \to BSF$), where $BF$ classifies stable spherical fibrations (oriented for $BSF$).  A lot more is known than Adams knew.  In particular, he didn't yet have the Adams conjecture.  Rationally, $BF$ is a point.  At an odd prime $p$, $BF$ factors as $BJ\times BCokerJ$, and at $2$ there is a non-split fiber sequence $BCokerJ \to BSF \to BJ$.  The $J$-map at $p$ is best thought of as a map $BSpin\to BSF$ ($BO\simeq BSpin$ at $p>2$).  By the Atiyah-Bott-Shapiro orientation, the $J$-map $BSpin\to BSF$ factors through the classifying space $B(SF;kO)$ for $kO$-oriented spherical fibrations and, at any $p$, $B(SF;kO)$ splits as $BSpin\times BCokerJ$ (but BSpin is seen in
two pieces, one carrying the Wu classes, the other the rest of the Adams splitting). The intuition is that $BCokerJ$ and thus the unknown parts of the stable homotopy groups of spheres can be ignored, leaving the focus on the quite computable composite $BSpin \to BSF \to BJ$. This is too fast, and details are in Chapter V of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra''.  In ordinary mod $p$ cohomology, calculations are thoroughly understood but don't shed light on your question. They are also understood for $K$-theory, by work of Hodgkin and Snaith, and here the intuition that $Coker J$ can be ignored is made precise by Hodgkin's result that $\tilde K(BCokerJ)=0$.