You don't really need dbplyr to run insert queries as DBI::dbWriteTable() will do.
On second thought (editing my previous answer) you can use dplyr - dplyr::db_insert_into(). My apologies for confusion about where the db_insert_into() function lives.
In PostgreSQL context DBI sometimes struggles with schemas, so you might want to consider a little workaround with a temporary table and executing SQL code directly via DBI::dbSendQuery():
dbWriteTable(con, SQL('tmp_tbl'), a_rstats_data_frame, overwrite = T) # a temporary table
dbSendQuery(con, 'insert into cruel_and_unusual_schema.final_table select * from tmp_tbl;') #flip the temporary table over
dbSendQuery(con, 'drop table tmp_tbl;') #clean up...