I really don't know how companies and Linked In administrator roles work. I do know that with Facebook, I end up with some weird and non-intuitive use cases where I can't tell that I've crossed over between acting as myself, the everday user, vs. the admin for a group that I help manage - and that has caused a certain level of information bleed... not so much unethical stuff, just bad use cases for my specific ways of interacting with the system -- so it's not out of the question.
To find out more, you may want to set up a dummy company in Linked in, and make a few test accounts adminstrators - then you can see if there's crossed signal and any way for one administrator's actions to be visible to the other administrator... I find that sort of information leak much more likely than super stealthy tapping of your Linked In account - but then that's me thinking like an engineering manager - the most likely destructive use case is usually people being bad at large scale system design and ignorant. :) Bad actors are the zebras, not the horses.
How to get out of it? My approach would be -
1 - Consider separating your linked in activity - make a you-at-work account, and add that as an admin for your work's linked in. Then detach your regular account from the company. Go ahead and leave the company as part of your Linked in resume - but don't be an explicit admin in the org. It's totally reasonable to say to your employer that you want a work life separation.
2 - When booking an interview first book time in your work calendar for the interview - I also like to give myself some slack - I'll book 15 minutes before the interview and 30 minutes after to give myself time to context switch and to unpack the interview a bit before going back to work. After you booked your calendar at work, then agree to the interview time. It means that if the interview time shifts around, you have more legwork to do on your calendar... but it's worth it. I usually label them something ambiguous like "meeting" or "personal".
3 - If your company still rebooks their meetings to collide with your private meeting - reject them. You had a time booked for something else, large amounts of meeting thrash are not respectful. In most orgs I've worked in, a rebooking meeting should obey the calendar constraints of the individuals.
4 - Is there is one specific individual who keeps rebooking their meetings? - talk to them. Ask why all the meeting time thrash. Don't disclose why it is bad for you, it's simply bad in general, and it's fair for someone on the receiving end of multiple rebookings to ask for a better process. For example, perhaps you and this person can some commonly available times that the meeting booker can count on. Good for them - it's easier to book meetings, good for you - don't book those times for interviews.
5 - ++ to getting off Linked in for scheduling. I have never found it a limitation to book via other means - like by text or email.
Most of this is in the sphere of "annoying meeting handling" and not so much about spying prevention... but it makes it hard for this person to interfere with you - whether or not it's coincidence.
Personally, I find that coincidences happen more often than you'd think. Things that can seem diabolical can still be innocent and extremely annoying. With no proof, I would address is a problem with meeting booking and work to eliminated the 'random' and get your schedule respected - it shouldn't matter whether it's an interview, a dr's appointment, or something else important - you should be able to book time and not have it constantly clobbered by meeting thrash.