IMO yes it would change things.
FWIW think of LE if they were suspicious of a family member. Now that family has been cooperative or at least trying to appear cooperative. Suddenly a PI shows up hired by the family? I would guess that it would change the whole tone of the relationship with LE.
Now if the PI shows up and the family denys hiring them, but says they intend to cooperate with the PI, then LE may feel they have to outline how it changes things with them, perhaps even warning them about how they could be being used. For instance if the family tells LE about something, then LE has been trying to keep it confidential. But if they tell it to the PI (hired by the media) and he is supposed to tell the LE, it is more likely to get out into the pubic arena via media. It might even have merited a warning to the effect that giving info to the PI instead of to LE could be considered obstruction of justice.
And yes I know that the grocery trip got into the public arena. But I don't think that was released by the LE. I think that perhaps there was talk at the store that she had been in just before Lisa went missing. Then with LE showing up and asking for the video there would be more talk. I think that got to media and media went to the store asking for more info and trying to determine what LE was looking for. And they found the video.