Will it be a good idea to take 3 months off after PhD defense to improve my skillset and work on my mental wellbeing?
I took about 3 months off. I just did not very actively search for positions and when I found one ask for plenty of time to prepare (the start date was basically "whenever you can make it" from my employers point of view).
It was amazing. Yes, in that time I was in the process of interviewing (but I had not applied for more than this one job yet, that just was too good to not try) and then packed up my life and moved to another country. But I also met with friends, slept in, decluttered my stuff, data and brain, visited family and just took time to recover. 

Everyone so far has reacted very positively to this fact. My family and friends admit that it made me a more happy, balanced person. My colleagues and supervisors (old and new) encouraged me to take the time off and stated multiple times that this was an appropriate and good thing to do. Even the lady at the office where I had to sign up as unemployed completely understood the situation and promised me not to bother me for at least three months, as long as I kept making progess with at least one application.