This is most likely not true at all, and for the reasons, i will have to dip into the context of the games during which gladiatorial combats were staged:

The events that included gladiators fighting each other were called munera, and were up of several different parts (venatio, executions and gladiatorial fights, mostly). They were usually hosted by Roman adnministrative officials, such as Priests of the imperial cult, duumviri (mayors), etc. They were massively expensive and the incentive for the hosts to set them up despite the huge costs will be part of the explanation for crowd participation:

The munerari (hosts) threw these Events because there was a tremendous amount of Prestige, and consequentially influence and popularity, gained from them. The reason why crowds favored gladiatorial combats especially was specifically the fact that they could participate in the decision over life and death at the end of a fight (this decision process was called the missio). The missio was seen as the climax of the fight, actually. There is a great article by Egon Flaig on this, which i will give in the sources.

The gesture of the missio (thumbs up/down) was to be given by the munerarius himself, but in order to maintain and gain social standing, he had to at least act like he cared about the calls of the spectators. If he didn't he would not be seen as a good host. For example, Tiberius (i believe it was him) is criticized for not acting according to the people's will on the matter Tiberius‘ Son Drusus was publicly scorned for denying the missio too often (Tac. ann. 1,76,3); another example for this would be Claudius' alleged cruelty and bloddthirst during the missio (Suet. Claud. 34,1).

The deeper meaning behind this ritul of concordance is that gladiators were to be seen as representatives of roman virtues, such as bravery, courage and skill in battle. These 'virtutes' (virtues) were judged during the missio by host and spectators alike, and to disagree on this matter meant to disagree on values fundamentally important to the Roman society.

So, to sum up: The 'thumbs up/down' (what exactly the gestures were is still debated) was very much a thing and vital to the ritual of Gladiatorial combat as well as the munera themselves. And crowd participation was an integral part of this act, because it signified that host and spectators were on the same page when it came to core values of roman society.

Sources Flaig, Egon: Ritualisierte Politik: Zeichen, Gesten und Herrschaft im Alten Rom, Göttingen 2003.

Flaig, Egon: Roman Gladiatorial Games: Ritual and Political consensus, in Roth,R./Keller,J.(Eds): Roman by Integration, Portsmouth 2007, p. 83-92.

Mann, Christian: Um keinen Kranz, um das Leben kämpfen wir!“ Gladiatoren im Osten des Römischen Reiches und die Frage der Romanisierung, Berlin 2011

Edit: Fixed the tiberius Part.