There is more ambiguity here than might otherwise seem apparent. In large part because existing precedents for Senate trials on impeachment create a curious set of procedures and precedents, and it's hard to see any clear consistency on them especially when they impinge on a President. A stark example is that Judge Walter Nixon was impeached, convicted and removed for the offense of lying under oath. Whereas President Clinton was impeached and acquitted less than 20 years later for the same offense: lying under oath. There were many Senators who voted on both such impeachments, and many of those who voted to convict Nixon ultimately voted to acquit Clinton.
Now we can also look at the case of the impeachment of Justice Pickering in 1804. The House impeached him for public drunkenness. The Senate voted 19 to 7 to convict him. The Senate then voted again, this time 20 to 6, to remove him from the court.
This sets a certain precedent that impeachment and conviction in the Senate need not automatically remove the convicted from their position. How does this work with the imperative language of "shall" in Article 2, Section 4? The use of "shall" is used in the constitution for mandatory actions, with rather different language used for optional actions. There are potential resolutions to this, but frankly none of them have been tested. One resolution is that the Senate implicitly did not think this was a conviction for "treason, bribery, or other high crimes and misdemeanors", so that the imperative "shall" did not apply. Which sounds reasonable, until you ask under what constitutional provision he was validly impeached in the first place? Clinton's impeachment lawyers would have argued it was the "good behavior" clause describing the term of Justices and federal judges. But why insert this non-explicit back door form of removal with no particular safeguards or protections on its use while also going to pains to enumerate impeachment and all of its safeguards and protections, without mentioning any link between them at all?
Senate precedents are also that the "barred from holding public office" stuff is also done with a separate vote, and only needs a majority: the two-thirds requirement being only applicable to the conviction.
Now imagine that the Senate does much the same thing in a Presidential impeachment trial. They first vote to convict. Supposing they do convict, they then follow their precedent and vote to remove him from office. What does the meaning of that vote become? Imagine in particular that a number of Senators decide that either they don't want to remove the President ("it'd be bad for the nation", which was in fact exactly the rationale given by at least Senator Byrd for why he voted to acquit Clinton despite being convinced he had committed high crimes or misdemeanors, and so likely a rationale used by many acquit voters), or that the very vote is itself unconstitutional because the imperative "shall" has already applied and removed him from office, and so they vote no or abstain, resulting in the vote to remove failing to get even a majority?
Now you're in a dire crisis situation where no one even knows who the President really is. The only hope at that point may be for the Judiciary (SCOTUS in particular) to decide to be decidedly un-Judiciary-like and make a rapid decision on the matter. And by rapid I'm thinking hours or less. What kind of damage could the country undergo when there's uncertainty, and perhaps active conflict, over who holds the power of the Presidency, for days, weeks, or months?