No, it is not the case that only one in ten reported rapes lead to a prosecution.
Only one in ten lead to a prosecution for rape. If we include the number that lead to a related prosecution (e.g. sexual assault) the figure is much higher, otherwise we wouldn't have 14% of cases leading to a conviction.
As for the accusation that the CPS cherry picks, there is some truth in that but it cherry picks for all crimes, not just for rape. The CPS is supposed to prosecute only where it believes there is a realistic chance of success. This does, of course, lead to a problem. Imagine two CPS lawyers each with 100 cases of all kinds (burglary, murder, rape, etc.). One charges all of them and secures 60 convictions. The other charges 10 of them and gets 10 convictions. In terms of the way the CPS is measured the second lawyer is the better one, but victims and the police would prefer the first lawyer. So yes, the CPS does cherry pick but, as far as I am aware, the evidence suggests that the proportion of rape cases referred to the CPS which lead to a prosecution is similar to that for other major crimes.
Between half and two thirds of reported rapes never get as far as the CPS. The most common cause for a case not getting that far is that the victim withdraws the allegation. The proportion of cases leading to a conviction is not dramatically lower than for other major crimes, although it clearly needs to be higher.
I am not suggesting we should be complacent and accept the current situation. Any rapist who gets away with it is one too many. The fact that so few rapes are reported at all is a problem (around 85% of rapes are never reported to the police), as is the fact that so many victims withdraw their allegations. But we will only improve things if we deal with facts, otherwise we end up trying to fix the wrong problem.