Again, it was not his job to do, he is not authorized by the league to do that, here it is according to the NFL Rulebook
Section 1, Article 8 of the NFL rule book
Non-player personnel of a club (e.g., management personnel, coaches, trainers, equipment personnel) are prohibited from making unnecessary physical contact with or directing abusive, threatening, or insulting language or gestures at opponents, game officials, or representatives of the League.
https://www.si.com/nfl/2023/12/04/dom-disandro-dre-greenlaw-sideline-example
And yes, he was escalating the situation, wasn't just my opinion, but also of the NFL Senior Vice President of Officiating as evidenced by the NFL Pool Report
"And we saw video that showed that there was a staff person that made contact with the player and that the player involved, No. 57, made contact with the staff member which warranted his disqualification. The non-player, he was contributing to that escalation."
https://sports.yahoo.com/ref-explains-greenlaw-ejection-49ers-023852821.html