The Greek word for 'love' in John 5:20 is phileō but it could be a mistake to put any emphasis on the idea of 'kiss', with this particular type of love, and especially not in this particular verse. Just look at the other occurrences of phileō in John's gospel:
11:3 - Lord, behold he whom thou lovest is sick... [Lazarus]
11:36 - Then said the Jews, Behold how he loved... [this one, Lazarus]
12:25 - He that loveth his life shall lose it...
15:19 - The world would love his own...
16:27 - For the Father himself loveth you, because ye have loved me...
20: 2 - ...to the other disciple whom Jesus loved
21:15-17 Lord thou knowest that I love thee.
This type of love is more about that between friends, especially in family relationships. Don't forget either that when Judas Iscariot kissed Jesus, that was not a kiss of love, but of betrayal. Nevertheless, the Greek for 'kiss' in Luke 22:47 & 48 (where Jesus asks Judas if he would betray him with a kiss) uses phileō in the sense of 'to kiss, be friendly'. However, the text in question is speaking of phileō as that of the Father towards the only-begotten Son, an on-going love, not a one-off occasion where a kiss might be indicated. The text cannot imply a kind of love where the Father would kiss the Son. Indeed, the Father commands sinners to kiss the Son (Psalm 2).
The other examples you give in the New Testament (John 16:27, 20:2, 1 John 3:1) do not use either of the two Greek words for a kind of friendship love that involves kissing.
Certainly, Jesus' claim to the men who hated him, that the Father loves him (5:20), riled them, but due to what the rest of that verse has Jesus telling them:
"...and sheweth him all things that himself doeth: and he will shew
him greater works than these, that you may marvel. For as the Father
raiseth up the dead, and quickeneth them, even so the Son quickenth
whom he will. For the Father judgeth no man, but has committed all
judgment unto the Son, that all should honour the Son even as they
honour the Father. He that honoureth not the Son honoureth not the
Father which hath sent him." (20-23)
So, when your comments include the statement, "One might even conclude the attempt to kill Jesus was exacerbated by 'for the Father loves/kisses the Son'..." the inclusion of 'kisses' is wrong. The attempt to kill Jesus was exacerbated by Jesus continuing that sentence to list ways in which the Father's love for him was exclusive, and to a degree that went far beyond mere friendship. If Jesus had 'merely' being speaking of a friendship love (which other humans could experience), he would never have gone on to state the Father's love shown in those exclusive ways.
It was almost as if Jesus wanted to enrage his angry accusers to the uttermost by beginning with a phileō love, but immediately progressing towards supernatural evidences of just how great that love of the Father was for the Son. Therein lies the astounding proof of the equal love of Father and Son, in a relationship that is so unique as to speak of the deity of both - in one divine nature, with absolute unity of the Spirit in that nature.
Therefore, the answer to your question is that it was Jesus' entire sentence in verses 20 to 23, which started with mention of phileō love but then made claims of a divine relationship between Father and Son that showed equality with His Father.
Source: Young's Concordance, 8th edition