No, the chain of custody starts when the evidence is collected, specifically from where and by what means without that there is no way to establish a CoC.

With these kits there is no way to verify that the DNA swab came from rape victims body or someone’s toothbrush.

This is the problem with these kits, any use of such kit would require a forensic nurse observing the collection, and that each swab is sealed in an evidence tamper bag and then the chain of custody can be established from that point on.

This is no different than any other physical evidence it either needs to be collected from the scene or forensically proven to be connected to it if it was recovered later or by someone else, and in cases when it does the defense will and quite likely successfully get it thrown out.