The grammar certainly joins the return of Christ to the Christians alive at the time of that event being gathered to him. It is clear from the construction of verse 1 that when Christ returns, then Christians will be gathered to him. Not before. There is no problem with the grammar.
Verse 3 substantiates that for Paul shows that a falling away (apostasy) from the faith must happen first and "the man of sin revealed". From what Paul wrote, it's clear that neither event had yet happened.
Verse 2 is really where some difficulty lies with its reference to "the day of the Lord" having come. Different people take that to mean different things, and contradictory doctrines about when events will happen, and in what order, have arisen. There are those who teach "a secret rapture", but not on the basis of anything in the letter to the Thessalonians. They have interpreted Revelation 4:1 to mean that somewhere between the end of Revelation chapter 3 and the 4th verse of chapter 4, a rapture involving a secret return of Christ and the glorified resurrection of the bodies of myriads of saints took place, but secretly. This view started to circulate in some Protestant groups from the 1930s onward.
However, if just the text of 2 Thessalonians 2:1-2 (actually, to verse 3, I would suggest) is considered, the construction is simple: The coming of Christ means Christians being gathered to him, but that "day of the Lord" won't happen until first an apostate 'falling away' happens, and 'the man of sin, the son of perdition' is revealed.
In every generation of Christians, the exhortation applies: to look for that gloriously spectacular return of Christ in faith that it could happen suddenly, while they are alive, but that if it doesn't, the next generation of Christians will hold to the same faith, which will enable them to endure all and any persecution, no matter how long Christ's return seems to take, from our point of view.
The book of Revelation really goes into that in much more detail, showing a time of massive apostasy and terrible deceptions luring mankind away from God and Christ. But at God's appointed hour, Christ will suddenly appear, all his deceased believers already with him, and those alive on earth will be caught up to join them. Immediately there follows the fleeing of heaven and earth from the great white throne, and him who sits on it - no place being found for them - and the Day of Judgment begins. (Rev. 20:8-15, which harmonises perfectly with 2 Thessalonians, the whole of chapter 2.)
To answer the question: Paul is saying the rapture happens instantly AT the day of the Lord. This is confirmed in what he says in 1 Thessalonians 4:13-5:3.