DNA will only connect to living relatives
Sorry I do not understand why you think that.
Any tissue from a person whether living or dead will/may contain their DNA even envelopes and stamps that have been licked may carry their DNA (I say may as it does deteriorate in certain circumstances or it can be contanimated with someone else's DNA). Hair, skin nails etc all contain DNA as do all bodily fluids etc.
It is perfectly possible to extract DNA from hair bloodstain and dried saliva from a deceased person, but we have to be realistic. One could be lucky and get a good DNA sample from the glue on the first envelope tested or one may be unlucky and have to have a number of envelope seals tested before gaining a viable sample.
For instance I have a number of envelopes licked and sealed by my mother. Such samples could prove a DNA link between my mother and myself or even my mother and my 1st cousin. In this case I already have the paper trail link, so the DNA would simply be additional proof but it would also be beneficial in providing my mother's full DNA profile possibly helping to connect more distant lines of the family.
This could be beneficial as I of course only have approximately 50% of my mothers DNA so having her full DNA could show/prove further distant cousin connections.
Cheers
Guy