I saw a video in which someone advised against removing all guitar strings at once and instead recommended replacing them one by one to avoid sudden changes in neck tension.
However, in another video from a reputable guitar tech, I saw them simply cut all the strings on an expensive Fender and replace them all at once.
Now I’m confused. Is there a clear advantage to replacing strings one by one over doing it all at once?
This is an old school thing. Not really relevant AFAIK on a modern bridge like Fender’s or MM or S-MM.
It’s crucial on a floating bridge like the old style Ricky or some of the semi-hollow bassrs.
Yup - you can clean the frets and oil that poor, dry, sad fretboard (if not sealed maple).
Loosen each string and take off one at a time.
Do not snip them under tension.
The one at a time thing is nonsense for most modern instruments.
Maybe … maybe… an old vintage/antique acoustic guitar, but not what we all are playing.
Nah…I use low-tack masking tape to mark around where the bridge belongs. Ask me how I started that practice… 
What you don’t want to do to a neck is to store it without strings and tension. The neck designed to be under tension to be straight-ish. That said, it’s very strong and in the general sense pretty indestructible and not as fragile as some would put it.
You can cut all at once or loosen the string then cut one at a time, it doesn’t really mean anything. I treated some of the cheaper bass poorly a few times but in general I take great care and replace them one by one. Not so much at the fear of damaging the neck but more of a string management cutting them and removing strings all at once may sound efficient but it’s not unless you have a big trash bin that you can dump all in safely. I usually coil them up one by one insert into the sleeve or old package so that it would not dangle everywhere and poke someone (my kids) in the eye, lol.
I pretty do what @John_E is doing. Treating instrument with care and respect is a great practice I try to treat my $50 bass the same way as I treat my $5000 bass.