There’s another reason why Firefox Quantum should have the previous/old search bar: when in a certain moment you want to make a few searches and you also need autocompletion from a particular search site that you usually won’t set as your favorite one (e.g. Wikipedia, WordReference, Youtube): it was better and faster to be able to change the default search engine with 2 clicks (so I’ll have the autocompletion from that search site instead) than having to enter the search configuration to change it all the times I need so (this is a big waste of time). Also it’s was way better to have the name of the default search engine shown in the search bar because making all those changes makes you forget which one is the browser’s current default search engine.
My suggestion to improve the search bar is this one: revert to the appearance and functions of the previous/old search bar and give the right-click button the less-desired options “Go to this [search engine’s] homepage” and “Search in new tab”.
And about losing the option of sending your search to a different site without making that site your default search engine (one-time search), my guess is that few people will miss this function because it’s not so common to do only a search (or even only two consecutive searches) within certain engine (maybe except Google or your favorite search engine).