Is a DNA test really worthwhile ?

[melba_schmelba]

I don't exclude DNA but it only connects to living relatives.

Other than what Guy has said (which is probably fairly unusual), many DNA kits are of people that have since died, probably going back to the early 2000s.

[Maiden Stone]

     Uncle George was born in 1889 when Sarah was 17. His birth cert is blank where father's name should be except for a 'G' struck out. Evidently she wasn't sure who the father was. Then 1891 there is a son Thomas Edward (both her brothers names). Father is just a line. 1893 a daughter registered in her maiden name. All these born in Nottingdale, a notorious slum. 
     If the family has a sordid past it makes it more interesting.

Change to registration of births in England 1874. A woman not married to the father of her child couldn't have father's name included on birth certificate unless he was present when the birth was registered or had given written permission.
Sarah may or may not have been able to identify father/s.
Another assumption you seem to be making is that women who lived in slums and had illegitimate children were prostitutes. 

[Geoff]

A DNA test found me my 2x Great Grandfather that was unknown to any one in our family, even his daughter, my Great Grandmother. She always told us her father had died when she was a baby and all she knew was that his name was Henry.  The only Henry that connects to the family was her uncle, her mother's brother but there was never any connection that he was the father in question.  My results showed that her father and my 2X grt grandfather was a chap that lived a few doors or streets away in Warbleton, Sussex.  For me, just that reason alone was all the proof I needed to justify buying a DNA Test kit. Regards Geoff