This would catch the majority of criminals, especially rapists.
No, no it wouldn't.
Here's a study titled The Role and Impact of Forensic Evidence in the Criminal Justice Process, which has some pretty interesting finds.
On page 95 you can find crime scene evidence for rape cases. DNA was found in 4.7% of cases, that's not a whole lot.
It is later stated that
As was statedthroughout this chapter, the vast majority of sexual assaults were committed by someone known to the victim. In these cases, identification is not the key issue because the defense will probably claim that the sexual activity was consensual. The immediate value of DNA evidence in these cases is minimal
Some more snippets:
The most frequently collected, submitted and examined forms of evidence were fingerprints, firearms and biological (blood and semen). For the sites included in this study and for the time period reviewed, DNA testing was rarely performed across all offenses and was concentrated in homicides and, to a lesser extent, rapes.
Very few reported crime incidents had forensic evidence that linked a suspect to the crime scene and/or victim (~2% of all cases, 6% of cases with crime scene evidence, and 12% of cases with examined evidence.) In terms of examined evidence, however, those percentages elevate to x% and y%. However, the conviction rate for the cases with linking forensic evidence was significantly higher than cases without such evidence. Furthermore, conviction rates were higher for offenses with two or more forms of individualizing evidence that associated offenders with crime scenes.
While forensic laboratories included in the current study did not analyze biological evidence and stains on a consistent basis, Appendix A shows that their use of DNA testing in such cases has grown substantially since 2003. While DNA testing procedures has the potential to individualize evidence and to link offenders to crime scenes and victims (Beaver, 2010), the present study has shown that even cases with strong forensic evidence are subject to investigator and prosecutor screening that assess the credibility of the victim and are amenable to defenses, including consent, that render DNA evidence less dispositive. It may be that stranger property offenses and those where ‘touch DNA’ is present, may constitute the offense category where DNA will have its major impact in the future.
Research may show that the discriminating and individualizing power of DNA evidence has its greatest impact on property crimes and those offenses with high percentages of stranger offenders. Personal crimes of violence and particularly those committed between acquaintances, intimates and family members should be carefully evaluated prior to forensic analysis, because in those offenses DNA individualizing evidence may have limited benefits.
Then there's the entirety of Appendix A with the title of;