At her engagement events, Letizia came across as overconfident and bossy when dealing with Mama's boy Felipe, taking the lead in the conversation where among other things she described their life together as a 'project'. She didn't realize when the Kings walked into the room and kept talking and talking, Sofia's facial expression said it all. I found the interviews not very well advised because Letizia was talking very confidently about something she clearly did not understand at this stage, how royal life would be, especially the dealing with the tabloid press. Little did she know back then, like still talking about leaving her journalist life slowly instead of instantly, while a few days later she had to terminate her contract with TVE.
She clearly made herself a target that day, and the royal house did nothing to help her. Letizia was put in her place and transformed into an accessory without a voice or an agenda for many years, enduring all the gossip stories that the royal house probably found rather helpful because they deflected from their own issues. Letizia very likely imagined her life as the Princess of Asturias certainly not the way it turned out to be, as Felipe's shadow, but had a far more active role in mind, a la Maxima.
But the Kings were already tired from fighting off Felipe's previous unsuitable girlfriends, especially Eva Sannum, and he was getting older and older, otherwise they would never have allowed their son to marry a divorced woman with a past. Sofia later said in an interview, if your child wants to marry an unsuitable person, all you can do is help to make the marriage work. I think all the camaraderie of the early days was kind of fake and many years later we saw at the mass that the relationship was probably never easy.
Letizia had all features of a career woman but married into an institution that in my opinion did not suit her character at all. She could have been an asset, but the powers that be decided against her and changing anything for her, and obviously Felipe was unable or unwilling to do anything about it. Actually, to this day very little has changed, it's all about Felipe and in comparison as a Queen Letizia hardly has any high profile events on her own agenda.
Therefore I don't believe that Letizia's popularity or unpopularity will play any significant role in the survival of the institution.