Why does BETWEEN use a btree index but "element is contained by" range operator (<@) does not?
An index can only support operators that belong to its operator class.
SELECT ao.amoplefttype::regtype,
       op.oprname,
       ao.amoprighttype::regtype
FROM pg_opfamily AS of
   JOIN pg_am AS am ON of.opfmethod = am.oid
   JOIN pg_amop AS ao ON of.oid = ao.amopfamily
   JOIN pg_operator AS op ON ao.amopopr = op.oid
WHERE am.amname = 'btree'
  AND ao.amoplefttype = 'timestamptz'::regtype;

       amoplefttype       | oprname |        amoprighttype        
--------------------------+---------+-----------------------------
 timestamp with time zone | &lt;       | date
 timestamp with time zone | &lt;=      | date
 timestamp with time zone | =       | date
 timestamp with time zone | &gt;=      | date
 timestamp with time zone | &gt;       | date
 timestamp with time zone | &lt;       | timestamp without time zone
 timestamp with time zone | &lt;=      | timestamp without time zone
 timestamp with time zone | =       | timestamp without time zone
 timestamp with time zone | &gt;=      | timestamp without time zone
 timestamp with time zone | &gt;       | timestamp without time zone
 timestamp with time zone | &lt;       | timestamp with time zone
 timestamp with time zone | &lt;=      | timestamp with time zone
 timestamp with time zone | =       | timestamp with time zone
 timestamp with time zone | &gt;=      | timestamp with time zone
 timestamp with time zone | &gt;       | timestamp with time zone
(15 rows)

There is no &lt;@ operator in there, so a B-tree index cannot support this operator.
GIN indexes can support &lt;@, but not with the constant on the right side.
You will have to rewrite the query and use BETWEEN.
Remark: There is not a fundamental reason why it is like that, it is just how indexes work in PostgreSQL. It is even possible to write an optimizer support function that would do exactly that, but your use case is so exotic that PostgreSQL doesn't want to spend optimizer time and development effort on it.