CLAIM: In 133 BC, the republic started illegally seizing land from the rich and redistributing it to the poor.
I would argue that this is a mischaracterization of the dispute.
The lands seized from defeated powers were almost always treated as public lands, and were then leased for income to wealthy Romans.
One problem with this practice was that it meant that Rome's public lands in effect became the property of only those Romans who were cash-rich enough to lease them.
Another property with this practice was that it favored system insiders who manipulated the leasing system to get the best lands for themselves at favorable rates.
The land reforms proposed by the Gracchi would have saved the Republic if they had actually been successful. After the fall of Rome, Byzantium solved its land-ownership problems by breaking the backs of the landlord class and undertaking successive land reforms not very different from what the Gracchi proposed - and Constantinople stood for 1000 years more than Rome as a result.
I think Beck is blinded by the "poor people vs. rich people" aspect of the public lands dispute, and immediately jumps in on the side of the aristocracy by reflex. Taking the Gracchi's side would actually be even more useful to the just-so story he's trying to tell - it would be very, very easy to depict the enemies of the Gracchi as profiteering politicians trying to loot the public lands for themselves, in the manner of machine politicians of the modern era.
CLAIM: Octavian refused to be called Caesar.
I think this is a fairly harmless error on Beck's part. I think he meant to say that Augustus never called himself Emperor. His point appears to be that Augustus was an emperor who never called himself one, and founded an imperial dynasty while pretending to restore Republican forms. And that's certainly true. The political value to Octavian of the Caesar cognomen is a little too inside-baseball for Beck.