Why does vacuum analyze change query plan while analyze does not?
Analyze is responsible for collecting statistics, and analyze have effects on selection of query plan and plan costs.
Indexes do not contain visibility information about the tuples. Because of that, reading data pages and filtering on resultset is also applied event though all columns in select is in the index. If all tuples visible in a page, Postgres does not need additional filtering operations. Postgres uses visibility map (vm) to achieve that.
Vacuum uses and also updates visibility map while reclaiming dead tuples. Because of that, vacuum changes query plan for using index-only scan if possible.