It may surprise (or offend) others whose first language is English, but "both" can refer to more than two word. It is, in fact, standard English to use "both" with any number of words in a coordinated list. So while it may may feel odd to some, the posted sentence is correct. For example, in the famous English poem "The Rime of the Ancient Mariner" (1834), Samuel Taylor Coleridge writes this: _______________ Farewell, farewell! but this I tell To thee, thou Wedding-Guest! He prayeth well, who loveth well Both man and bird and beast. He prayeth best, who loveth best All things both great and small; For the dear God who loveth us, He made and loveth all. ______________ Here we see "both" applied not only to two words (i.e., "both great and small") but also to three words (i.e., "Both man and bird and beast"). In fact, Merriam-Webster dictionary currently cites Coleridge as an example illustrating that the word "both" may apply to any number of coordinated words.

"I have appreciated his precision, reliability and punctuality when it comes to meeting the agreed deadlines" If you use "both" you have to refer to two things. In this case you are referring to 3. That's why I deleted it

It is not correct. You use the word “both”, but then list three attributes. In English, the word “both” can only be used to refer to two things. For example: “I admire both his bravery and his intelligence.” If you want to list more than two things, you can simply list them: “I appreciate his kindness, wisdom, and generosity.”