There were 3 possible outcomes here:
1 - 3rd party involvement, kidnap, murder
2 - Suicide
3 - She has got to the point where she feels her family are better off without her and wants to disappear and not be found again
Everyone, including the police, were hoping that is 3. Then they can still find her alive and get her the support she needs and bring her back home.
They released the information to help find her
If she didn't want to be found - she could have been anywhere in the country - very easy for a person to travel without having to buy tickets - hop on and off trains, hitch on the back of trucks etc etc. And she will changed her appearance so she is not easily recognised anymore and she certainly wont be using her real name.
When we were told she has alcohol issues - it could be that a shop owner in say Aberdeen has had a female come and go from his shop buying alcohol over recent days with loose change and shop owner has not really had a good look at her. But now they are aware she has alcohol issues - they will give her a good look next time or talk to her. Or support services she might be accessing under a pseudonym, drop in centres etc
She also has to eat - so she may have been accessing homeless feeding stations to get food, or finding food in restaurant bins etc etc. and she will be keeping her head down doing this to keep a low profile - the proprietors might give her a second look next time she attends. and talk to her, look at her etc phone the police if they think it might be her.
The police will have had this information from the start - but probably did not want to release it in the hope that she would have been found sooner. It got to the stage now where it needed to be released to widen the public's gaze for her.
Ask yourself - how many homeless people have you seen in your area recently huddled in a shop door way. Have you gone over and given them a good look over to make sure its not a local missing person? If you haven't done this, then most other people won't have done it either.