I would say that the Spirit was driving the church outwards to declare what God had done for the world through Israel - not simply what God had done for Israel through Jesus. It’s a subtle, but important difference. The relevance of what God had done for Israel was for the world - not Israel alone. In that sense, the focus of the prophetic community was the world, not Israel.

I’ve also argued that the impact of this message on the gentiles happens  before Acts 10 - with the founding and development of a mixed Jewish/Gentile church at Rome (probably by Jews/Jewish proselytes following Pentecost - Acts 2:10-11), and the bearing of the message to Antioch - which according to Acts 11:19, happened after the stoning of Stephen (in Acts 7).

I don’t see any division between what Peter preaches at Pentecost, and what is proclaimed throughout Acts to the Jewish or Gentile world. Again, contexts inform interpretations. Peter’s message at Pentecost was directed to Jews - so he emphasised the significance of that event, and the resurrection of Jesus, in terms of its relevance to Jewish history. But the rest of Acts goes on to develop the significance of that Jewish history for the rest of the world (as in Acts 17:22-31). The significance of Pentecost was that the life of God, the Spirit, had been given where previously death had been the ultimate reality and experience. Jesus was ‘Lord’, in the sense that he had overcome death, and had dispensed life on all who believed in him. The worldwide significance of this message is very obvious.

The OT has different angles on the meaning of ‘the day of the Lord’ - but whatever it was, it was not fulfilled in its entirety with judgement on Israel or Rome in the 1st/2nd century. It’s not a question of it being ‘the end of history’, but a question of it being the end of a certain kind of history - in particular, the end of ‘this evil age’ and the beginning of ‘the age to come’. The unexpected development of the coming of Jesus was that ‘the age to come’ appeared in the midst of ‘this evil age’ and so continues to this day. When ‘the day of the Lord’ finally comes in its entirety, in a way which will include all that is implied in its OT useage, it will be the entire end of one phase of history (‘this evil age’), and the entire fulfilment of ‘the age to come’ for creation in all aspects.