Hi, this is feedback I get a lot too so don't know how much help I will be buuuttt...
I think basically the 'lack of strategic thinking' feedback/impression that people get of you is quite common where you reach a point in your career where you have a lot (maybe too much sometimes!) of practical, technical, operational experience but haven't yet been in a senior post and/or demonstrated leadership. You have to remember at interview stage they aren't just assessing whether you can do the job (they obviously think you can, but so can all the other candidates), but also would they like working with you, so I do think that feedback can be basically a bit about your personality and how you are coming across. Certainly for me I know I can give the impression I am a super-detail focused person who loves to get a process completely right and efficient from start to finish - you'd think how great, who wouldn't want that in their team, but it can really irk certain people. I think it makes them worried I'd be pedantic/anxious about ensuring everything is technically correct and unable to see the bigger picture and I'm guessing they also think (and probably correctly TBH!) that I'd be a pain in their backside about always wanting to make improvements, pointing out problems and (and this part is incorrect) perhaps not keen on change or unable to be pragmatic and problem-solve/work around where needed. I also think maybe I can come across as lacking in ambition (for the job role, not in my career) compared to other candidates because I focus too much on practical realities (it does then really quite irk me when I have an experience of someone who talks all the talk but has no idea how to practically achieve any of it gets promoted ahead of me AND then fails to get their comeupannce but hey ho, life isn't fair) .
What is the solution, many won't agree but I tend to do a bit of 'if you can't beat them join them' - simple things like: using all the current favoured buzzwords and corporate bullsh*t, much as it makes me feel like vom-ing using that instead of simple, practical english, showing you understand their company's ambition and share that goal (with tangibles on how to contribute to that even if you are applying for a low level role), having really good examples of work at a higher level that is more proactive, rather than every example being how you've managed a problem that's come up or completed a task you were asked to do, really sell these as how you're 'driven transformational change' (puke) aka implemented some process improvements or a new service/product or something and 'facilitated innovation' (urgh) aka had a creative or new idea or taken someone else's idea and followed it through.
Basically what business people tend to term 'strategy' or strategic leadership really just means choosing an end goal or end state, whether that's to make more money or bring in more clients or improve the quality of your service or whatever, identified the correct steps in order to achieve that goal (which may involve an element of creativity or 'outside the box thinking', as above be careful not to get yourself pegged as someone that just always does things 'correctly' because that's how they've always been done), then made a realistic/achievable plan to do those steps, a awful lot of guff gets talked but it's fundamentally that simple and logical at the end of the day. So I do find it helps to structure answers in that way even of quite small examples of things I think I have done to genuinely make a difference.
Hope that ramble makes some kind of sense!