I've made the same and, of course:
2. Opened KLAX ADE file
This is probably the first mistake: if with "KLAX ADE" you mean the ADE save file, which is usually named with an ".AD2" extension, it's likely THIS file can't be used anymore, because it was already compromised by editing with a previous version that HAD a problem editing/removing custom jetways.
You should have started using the "Open from .BGL" command in ADE, from the ORIGINAL (unmodified) KLAX_AP.BGL that comes with KLAX! To get this, run the KLAX installer again, so it will be restored.
3. Deleted parking spot #106
Not sure if you are referring to the ADE #106 Index or the Parking spot #106 number. Shouldn't make any difference but, in any case, I've removed the Parking #106, which was I believe at Index #63.
OF COURSE, as I've said in my previous message, I ALSO removed the now orphaned jetway for the removed parking. Not sure what happens, but you surely don't want to leave an orphaned jetway, when they are supposed to be associated to a parking.
4. Clicked on "Compile Airport" and selected the ADE file with the same name. ADE said "Failed to generate jetways' binary"
Not sure what you mean here: if with "ADE file with the same name", you mean the .AD2 file, than it's obviously wrong: since you were trying to overwrite over the ADE file itself. I hope you meant the KLAX_AP.BGL, which is not an ADE File, it's the KLAX AFCAD.
In any case, I did the same (overwriting the KLAX_AP.BGL, of course) and ADE compiled without errors.
5. Reloaded FSX and went to KLAX. All jetways (except for some at TBIT) are gone.
Did the same and, of course all jetways at the airport are fine, except the one for the parking I've removed, which is exactly what is supposed to happen.
It seems like your jetways are created with default FSX codes, therefore it's total pain in the butt when it comes to parking spots deletion
It seems you are still doing something wrong, so you are mistakenly assuming KLAX has a problem being edited (with ADE, other editors like AFX never had a problem to begin with, even with older versions), when it clearly hasn't.
I repeat and confirm, again, there are no problem whatsoever editing KLAX parking spots with custom jetways, KLAX never had a problem in the first place, and ADE can handle it without any issues.