I think it's even simpler than that. A lot of the people conflating anti-Israel sentiment with anti Jewish sentiment are ethno nationalists. On the Israeli side, those would be Jewish nationalists; but here in the US, the sentiment is disproportionately coming from Christian nationalists. Incidentally, these people also tend to be the same people who conflate anti-current-governing-coalition-and-policy-of-Isreal sentiment with anti Israel sentiment more generally, because that conflation is part of fascist ideology, and ethno nationalism tends to be a fascist ideology.

The reason we see pro-Zionist media ferment anti semitism is simply that the Zionist movement is ideologically aligned with most anti-semetic movements.