If you do not have a dagger marker, LU staff will claim (whether right or not!) that the ticket is not valid and each time you wish to pass through a LU barrier, you may have to argue with them!!!
This is what ATOC say
National Routeing Guide said:
The via London, Maltese Cross symbol on a ticket signifies that the ticket
may be used via London Underground or Thameslink services. Unless a ticket
specifies that the journey must be made via London, passengers are free to
use an alternative ‘permitted route’ for their journey as provided by the
Routeing Guide.
In some instances (particularly long distance cross country journeys) the
Manual will show an "any permitted" fare but without the via London, Maltese
cross symbol. Reference to Section C (the “yellow pages”) may show via
London to be a permitted route for this journey and in such instances travel
via London to include cross-London transfer would be permitted.
So the LU barrier drones are wrong!!
But, prepare for many arguments if you get such a ticket.
In 2006 a Grantham-Stroud ticket was routed Any Permitted without a dagger/maltese cross symbol, and yet was routed via London, and I had to argue at King's Cross until one of the barrier staff agreed with me and asked the other one to stop arguing with me. At Paddington I couldn't be bothered with another argument so I went directly between platforms avoiding barriers. GNER and FGW staff correctly accepted the ticket without question on outward and return journeys.
On another occasion I used a Not London ticket via London on a day when all Voyagers were taken out of service and passengers were instructed to go via London. I was challenged at the barrier by someone who was polite and admitted that they had not read their daily notices as they had just started their shift, and when I asked "have you not seen the news?" apologised and let me through.
No maltese cross doesn't automatically mean you are not valid via London but LU barrier staff are not really taught to think for themselves and are not given decent quality training and are instructed (or, "programmed"!) to refuse to accept them. This will likely result in an argument, which can only be won by being assertive and bringing documentary evidence supporting your case. (ie, not by being aggressive, but equally not by being passive either!)
I don't really blame the barrier staff, they are badly trained. That said, there are some very good ones who do use common sense and are much more familiar with ticketing than they are trained to be (e.g. Daniel!) but these are rare and won't stay in that job for long as they'll be on their way up through the ranks.