Did you try clicking the 3 dot menu to the right of your "other search engine" entry that you would like to make default?
For those of you who wear aluminum foil hats, the voices lie, don't believe them!
Mostly my replies are about Windows OS. If not I try to remember to specify Linux.
Personally I use Context Menu Search more than anything but I have a list of 5 in the default and a LOT in the "other" field, how did you end up with only one in the default as maybe that has something to do with the issue? Just trying to see what I can understand on how it got to this point in hopes it helps get things back and also offering a possible alternative.
I like Context Menu Search FAR more personally anyway as I can use it with all browsers and easily have all matching between the laptop and desktop and as stated other browsers as well. It is also so easy to have the option to search so many custom options not normally offered (like Newegg for example) with a right-click.
I deleted those because I did not want them there and want to put in some others instead.
I am not certain this is ideal but I think this would be the best option.
Reinstall the browser, probably after you remove it so they all return to be like it was originally as I am not positive how to get them back otherwise.
Get a text file listing all that you want to add and note what you want to be removed. Rather than removing them this time just edit the ones you want to remove and use the existing fields to add the ones you want to keep. Then they'll all be in the right places and if you prepared the list beforehand you can figure out the exact order you'd like them listed in as well.
Otherwise, you could just try the Context Menu Search extension for a better solution (I think).