Could it be that the OP is confusing ‘Insert mode’ with regular ‘Note input mode’?
Your comparison of a scorewriter to a typewriter is useless as you are missing one important aspect of music notation: the existence of rests.
In spoken language, hence in written text one word comes after another and you read it just like that. In music however a single note could be followed by a long period of silence before the next actual note takes place. Therefore a score writing application must be told at what rhythmic position the note you want to enter is to happen. This is what the rhythmic grid is for. Have you noticed the small yellow lines that appear above the staff when you are in note entry mode? They represent the grid. On the very bottom left corner of the Dorico window you can see what note value the grid is set to, default is 8th notes. When you use the mouse for note input you will notice that the grey shadow note will only snap to those positions. For keyboard note input you move the actual input position (the ‘caret’) with the arrow keys. If you want to input a note at a position 1 quarter note after the previous note you must tell Dorico to do so.
The benefit of this might seem non-obvious in your specific case, but notice that it gives you also the possibility to enter the next note not directly back to back but on any position, Dorico will automatically insert the necessary rest symbols. In your typewriter analogy you would have to type these rests manually.
You can actually input notes in typewriter-style when not using mouse input: When you input notes with the computer keyboard or a MIDI keyboard the caret will move exactly to the position after the note you just entered.