what are the best ways to write these so that the company will not be able to find out the person who wrote this particular feedback.
Confidentiality is not the same as Anonymity. The survey feedback may be collected confidentially, but only in some cases, it is partially anonymized for certain responses. In every engagement survey I've participated in, leadership at the director level or above could see who said what, though my immediate managers/ skips managers would not be able to see that.
But if someone leaves breadcrumbs of personal interactions, I don't think it would be hard for the manager to understand who might have been behind it.
That said, your original question is:
How effective are Employee Engagement Surveys? (Employee Perspective)
I've seen them to be effective at an aggregate level for the senior leadership to understand what they are doing wrong/ what can be improved at an organizational level. So, in a hypothetical example, if there are 20 of you under a leader, and 10 say the same thing needs to be improved, then that is something that gets attention. However, if only 1 person cribs among the 20, I would say unless the charges are very serious, that will be treated as an outlier rather than the trend.
Another way the surveys are useful is that when done on a periodic basis, they help leadership gauge the ups and downs in these trends. Continuing the hypothetical example, if for a question they got 10 negative response in previous run but now get only 5, this shows improvement. (and vice versa).
So as an employee, if you and others are voicing the same problems, you can expect some thought cycles to be spent on solving them.
However, to me, the crux of your question seems that you want to pass feedback regarding your manager, office politics back to the organization. If that is the case, I would suggest having a 1:1 with your skip level manager. Go with a documented list of malpractices which you can use as facts to back your opinions in this meeting. Most leaders keep their eyes open to such feedbacks, but if you still think you are not being heard / this is what the company culture is - you are left with 2 options - learn to live with it, or brush your resume.