The only programs I have that are putting things in the titlebar are putting tabs there, and those programs do not cause me notable problems. For me the positives are much stronger than the negatives.
document-name-dropdown, search-box, username, user-icon.
Elements that do not allow dragging the window, because they react to mouse-down, include:
application-icon, quick-access-toolbar, notifications-bell-icon, and the standard titlebar icons.
Maybe the addition of dedicated Move button next to the minimise, maxmimise and close button would be a reasonable compromise?
That would provide a consistent target to click on with your mouse. Obviously a button is a lot smaller than the entire titlebar for clicking on, so there would be some efficiency loss for people that regularly move windows (I am one of them). Getting move use out of the titlebar space would be worth the minor inconvenience of a more accurate click-drag to move operation.
This does become more problematic when there are so many tabs they become tiny and the whole title bar of the browser is taken up by tabs so that there's no dead-space to drag the entire browser window to another screen. I try to keep minimal contexts (due to personal brain capacity issues), so I don't run up against this very often - I manage links that I want to keep and go back to using other means, or if I forget them then they weren't very important in the first place.