Well yes and no. If you are proven then to an extent the club will give give more liberties to build your project. Take Klopp at Liverpool, sure they gave him time to build the project having had huge success at Dortmund. But also, he had more freedom because at the point of take over Liverpool was a very (or at best modesty) pressured club. There was no expectation to win anything instantly compared to Xavi coming in here or Nagelsmann going into Bayern.
But even so, Klopp is/was capped on signings (notorious example of him wanting Brandt over Salah) while if he were at Dortmund or Maine, he would have probably got his choice.
I'd say Arsenal would be in the same boat where they were a top level club in name but not expectation nor pressure. Klopp and Arteta get the can at Barca or Bayern for having the type of seasons they had at the beginning of their EPL tenures for example, at these respective clubs.
What I don't agree with is that there are clubs 'pressured' to win and clubs not.
Every big club's goal is to win titles, the difference is how you approach that objective.
Liverpool for instance, had a method and it involved trusting the right manager long-term. That doesn't mean he could do anything he wanted, hence the involvement of other departments in transfers for instance. And eventually it yielded phenomenal results, given their net spent over the last 8 years.
The bad seasons that you mention are part of the process, and you have to accept them as long as the direction is progressive and the roadmap is clear
Look at Arteta. He was never gonna win in his first 2 or even 3 years, but the signs were there: every 6 months there was progress in the way they were playing.
Same approach can very well apply to bayern and Barca if there is plan and method (questionable about both right now)
The whole mentality 'win now' 'big clubs should get titles immediately' is a Madrid (and Juventus) mentality that has infected many big clubs unfortunately