composition versus inversion

While trying to convey to an OP on X validated why the inversion method was not always the panacea in pseudo-random generation, I took the example of a mixture of K exponential distributions when K is very large, in order to impress (?) upon said OP that solving F(x)=u for such a closed-form cdf F was very costly even when using a state-of-the-art (?) inversion algorithm, like uniroot, since each step involves adding the K terms in the cdf. Selecting the component from the cumulative distribution function on the component proves to be quite fast since using the rather crude

x=rexp(1,lambda[1+sum(runif(1)>wes)])

brings a 100-fold improvement over

Q = function(u) uniroot((function(x) F(x) - u), lower = 0, 
    upper = qexp(.999,rate=min(la)))[1] #numerical tail quantile
x=Q(runif(1))

when K=10⁵, as shown by a benchmark call

         test elapsed
1       compo   0.057
2      Newton  45.736
3     uniroot   5.814

where Newton denotes a simple-minded Newton inversion. I wonder if there is a faster way to select the component in the mixture. Using a while loop starting from the most likely components proves to be much slower. And accept-reject solutions are invariably slow or fail to work with such a large number of components. Devroye’s Bible has a section (XIV.7.5) on simulating sums of variates from an infinite mixture distribution, but, for once,  nothing really helpful. And another section (IV.5) on series methods, where again I could not find a direct connection.