Disclaimer: I left academia for "industry" right after my PhD and never regretted it. I am still close friends with multiple people currently in French academia, so I am reasonably up-to-date with conditions there.
A job is a job
I will start by a very general piece of career advice.
"Job" means that you do things that are, according to the majority of the population, unpleasant to do. Society pays you a salary to motivate you to do it. The salary allows you to do stuff you like when you are not working.
A specific job can be less unpleasant for a specific person, and that’s part of what should guide your decisions beyond salary (which is more objective). Academia does have significant perks; I do not regret them, because I consider the drawbacks to be too big, but I understand that different persons might have made different decisions.
However, very few people who do a job they love, that is, that they would keep doing even if they kept the perks of doing it (salary, talking with people during the day) without doing the associated work. If anyone tells you that you should ignore a work-related problem (be it a low salary, a stressful work environment, sexual harassment, whatever) because you’re lucky to work in $DREAMJOB, they’re being manipulative and you should not listen to them.
Academia vs. industry: general considerations
The difference between academia and "industry" is smaller than imagined by people who are not familiar with both. The variation within "industry" is vastly greater than imagined by people who have only ever been in academia (hence the scare quotes).
In my opinion, the biggest difference by far is that in academia, past a certain stage of seniority (somewhere around PhD or postdoc), you have no boss. Yes, you might have a person who needs to approve your timesheets, sign off your purchases, and so on. But you do not have someone who requests frequent status updates on your work; requests that you change what you work on because something else came up and that’s what you’re doing next week; etc. You still have pressure (to publish, to find funding etc.) but it does not come from a single identified person.
Having no boss is a big plus for some people. There are many small companies of one to three people where the owner does their core job of expert foobarer, plus sales representative, plus accountant, plus HR, and they end up working 70h/week. They would make as much money in 40h/week at FooBarCorp as regional expert, but they would have a boss, and they don’t like that. (Typical example: a single-employee computer repair shop vs. the IT hardware department of a big company.) It’s possible to go through academia without knowing what a boss looks like, because the boss figures (advisors) are much more likely to be absent ("I will see you in two months for a ten-minute meeting"), peer-like ("I am not really your boss, more like a more experienced colleague"), or abusive ("if you fail to impress me, you are worthless as a person") than outside academia.
Your list
Thinking about complex things
a.k.a. "having a non-boring job". Absolutely feasible outside of academia. In fact, most companies that recruit PhD try to put them to work on stuff that requires thinking (otherwise, why recruit those people? the non-thinking crowd is usually cheaper).
Statistics, data analysis, simulating stuff
a.k.a. "job topic". Absolutely feasible outside of academia. You just need to not apply to accounting or HR positions.
Being able to follow my own ideas
That one’s more tricky, though. In R&D divisions, you might have a big research project one year, find promising results, and have funding cut to zero the next year. Even if the customer agrees that you had promising results, money fluctuates quickly.
You definitely cannot choose the details of what you work on, but I promise you that R&D positions including interesting research questions, not just "optimize some knob-turning for my machine".
Discussing my ideas with peers
Absolutely feasible outside academia. Possibly even more so, but it depends on the company/team culture, and it is hard to know that before recruitment.
Possibility of home office & flexible working hours
Absolutely feasible outside academia. Depends a lot on the company, but those are questions you can ask during the interview.
Managing
Well, that one’s tricky.
In my view, academia does not require much managing duties from you (yes, if you want to get to the top of the food chain, but you can be a perfectly competent and happy late-career tenured professor with a couple of PhD students and interns).
In industry, many companies require you to take on management duties as you get more experienced. (In my opinion, this is a stupid decision, and the other companies that allow an "expertise" track are much smarter, because not everyone wants to become a manager or can be good at it. But that view has not reached the majority of HR bigwigs. I left a company over this problem.) You can, kind-of, ask the question in interviews, but it can be hard to trust the answer.
bureaucracy, politics
Unfortunately, present in every human endeavour. Arguably moreso in academia than in industry, but it highly depends on the company.
asking for grants
Present in R&D as well. The grants are more generous than in academia, and the grant-hunting process is usually less bureaucratic, but still there.
Low salaries
Yeah, the private sector does a better job at those. (It can be a 20% bump or a 200% bump depending on your country, topic, etc., but it is never negligible.)
I am also currently tired of having to move so much
The private sector does do a better job at those. Once you are set near a big-ish city, in a place where you speak the language, you can usually do all your career there. (If you want to work in the countryside, that might limit your options more - but then, that limits your options to basically zero in academia.)
Suggestion
The most natural fit for any PhD/postdoc who still wants to do research stuff would be R&D positions. Large-ish companies have designated R&D subdivisions, where you might be the senior expert of blue FooBars in the FooBar department in the Foo division. In a small-ish company, you might be the anything-vaguely-physics-related person. (In my view, the former is much closer to academia than to the latter. Both are interesting in very different ways.)
If you look at mid-size companies (roughly between 20 and 200 people), don’t discount job offers just because it says "engineer" in the title. "Engineer" can be anything between "write up technical specifications and check legal compliance" (boooring) and "do research (but the HR software does not have "researcher" as a label)".
