You are reading the clause wrong. The senate, if they convict, “MAY” remove from office and “MAY” disqualify him from holding any office. A prime example is Rep. Hastings, (a former judge), who was convicted and removed but not excluded from holding “office”.
The house has no say in what punishment the senate can impose since the senate has sole authority to hold the trial and determine the punishment, (just as a prosecutor can’t force a judge to sentence the defendant to a specific punishment after a guilty finding, neither can the house).
True, but the Articles of Impeachment that the House passed do that. Both contain the sentence "President Trump thus warrants impeachment and trial, removal from office, and disqualification to hold and enjoy any office of honor, trust, or profit under the United States." So if convicted Trump would be barred from future elective federal office. The Hastings articles contained no such clause.
The house has no say in what punishment the senate can impose since the senate has sole authority to hold the trial and determine the punishment, (just as a prosecutor can’t force a judge to sentence the defendant to a specific punishment after a guilty finding, neither can the house).
But this is not a criminal trial. If the Senate votes to convict, and we all know they won't but if they did, then they basically are voting to accept the impeachment articles that the House sent them, sentence for sentence, clause for clause. And if those articles contain a ban on future office then that ban stands.