By adding

{
  ...
  "converters": "tinyint-one-to-boolean,jdbc-sink,duration-converter",
  "tinyint-one-to-boolean.type": "io.debezium.connector.binlog.converters.TinyIntOneToBooleanConverter",
  "tinyint-one-to-boolean.length.checker": "false",
  "jdbc-sink.type": "io.debezium.connector.binlog.converters.JdbcSinkDataTypesConverter",
  "tinyint-one-to-boolean.selector": "my_database.my_table.my_column,my_database.my_table.other_column",
  "jdbc-sink.selector.boolean": "my_database.my_table.my_column,my_database.my_table.other_column",
  ...
}

to my source connector config I was able to replicate the tinyint as boolean in PostgreSQL.

see also https://debezium.io/documentation/reference/stable/connectors/mysql.html#_tinyint1_to_boolean