Options for archiving old MySQL 5.7data
You need not move the data into another server just for performance improvement.
You could just move the data into another table (named sometable_20201023, assuming the archive/purge is happening on 2020/10/23 and you are OK with such a naming convention) and delete from the existing table. Since the original table is small, performance should improve.
Performance takes a hit when active data set does not fit in memory and disk I/Os are triggered. Having  inactive table(s) in your server will not cause any significant performance issue - your data dictionary size may go up by a bit.
In case you are concerned that your back-up/recovery time is also going up and want to fix it, this approach is not the best way. Moving the data to another server is best.