Postgres seach jsonb with indexes

Question

Im new to postgres jsonb operation. Im storing some data in Postgres with jsonb column, which has flexible metadata as below. I wanted to search different unique metadata (key:value pairs)

id, type, metadata
1, player, {"name": "john", "height": 180, "team": "xyz"}
2, game,   {"name": "afl", "members": 10, "team": "xyz"}

results should be something like below, distinct, order by asc. I wanted it to be efficient using some indexes.

key    | value
______________
height   180
members  10
name     alf
name     john
team     xyz

My solution below hit the index for search but sorting and distinct wont hit any indexes as they are processed values from jsonb.

CREATE INDEX metadata_jsonb_each_text_idx ON table
USING GIN (jsonb_pretty(metadata) gin_trgm_ops);

select distinct t, t.* 
from table u, jsonb_each_text(u.metadata) t 
where jsonb_pretty(u.metadata) like '%key%' 
order by t.key, t.value

Appreciate any thoughts on this issue. Thanks!