So day 1 was an exaggeration but yeah, the House could indict a president much earlier on if ultimately the point is to let the Senate keep the reigns on him for everything thereafter.
And in addition to the fact that this aught to be illegal, I don't think it makes sense as a scenario. With this House/Senate/Presidency, the same threat of conviction by the Senate is always there, because as soon as the Senate signals they'll convict Trump the House'll impeach Trump.
And also the fact that all the other plans to "reign Trump in" over the years led Senate Republicans to exactly where they were on Wednesday, with Trump sending rioters to their chamber, gallows erected outside and a neo-Viking sitting in Pence's chair. If they're not going to acquit again then I don't understand why they wouldn't just be done with Trump and convict him on this rather than something later in the week.