1 John contains the same ideas as the short letters but develops them more fully. It seems impossible to fully understand 2 & 3 John without also understanding 1 John.
So what was the purpose of these short letters?
1 John does not fit the format of a typical New Testament letter: it does not contain the typical greeting and it addressed to believers at large rather than one particular congregation or region. Only 1 John and the letter to the Hebrews begin without identifying the author by name or title and without any introductory greeting.
As a cover letter, 2 John supplies the typical general greeting missing 1 John and summarizes the themes of the letter. In 2 John 1:1, the “elder” is the Apostle John who was the last surviving apostle and an old man when he wrote this letter. The “chosen lady and her children” would be the community of believers and the specific local churches who received the letter.
If 2 John is the general cover letter to 1 John, 3 John seems to be a specific cover letter to an individual to introduce 1 John to a specific church or location.
3 John (1:9-10) implies that 1 John didn’t reach a particular location, so John resent the letter to an individual (Gaius) with a personal cover letter, so Gaius could introduce the letter into the area it missed.