I would appreciate an explanation that addresses the following questions, and hopefully offers any other related insights:
What you are referring to is vibrato, not tremolo.
All singers use some amount of vibrato.
These days the use of strong vibrato is mostly associated with opera singing, but this was not always the case.
Vibrato helps you sing louder, and with less fatigue to the voice. Some singers learn to sing with a little vibrato or a lot, and learn how to control how much.
In the Western tradition, Vibrato is essential to singing on stage if you are unamplified -- no microphones, no PA system, no speakers.
Singing with vibrato is something that many professional singers learned to do in the era before amplification was invented and became widespread and practical (in all the years before about 1925). The idea of singing with a band without vibrato, which used to be called crooning (that was originally a derisive term) only became practical if you assumed the use of a microphone and amplification on stage, or in the case of recording the singing voice in a recording studio.
The single person most associated with developing a new style of singing based around using a microphone, without vibrato, is the great Bing Crosby (1903-1977). This kind of singing, which most pop musicians use today, didn't really exist before Bing.
If you go back and listen to some of the earliest phonograph recordings, from the 1910s and 1920s into the 1930s, in most styles of pop music around the world, you will hear that most of those singers sound somewhat like today's opera singers. By the same token, singers in Broadway musicals used a style that incorporated some degree of operatic vibrato up until the mid-1960s, because up until that time, singers in Broadway musicals sang without microphones and amplification, while accompanied by a live orchestra, just like opera singers do today. If you don't believe me, listen to original Broadway cast recordings from before the 1960s.
Vibrato is essential to operatic singing because using vibrato is part of the technique which enables a solo singer to be heard, unamplified, singing with a 40 piece orchestra in front of an audience of 2,000 people. Using vibrato also enables a singer to sing very loud and project clearly for a considerable amount of time without fatiguing or injuring the voice. You can't do that without vibrato.
I have read accounts of pop singers in the modern era who, over years of singing, damaged their vocal cords from poor technique. After they undergo corrective surgery to try to get their voice back, they have to continue with rehabilitative therapy with a voice coach. The therapy includes learning to change their singing technique by incorporating some vibrato so that they will reduce the risk of injuring their voice all over again.
Of course some pop singers who do not use vibrato develop a permanently raspy, weak voice (think Tom Petty or Bob Dylan). They like it that way, and some way or another they learn how to keep using their voice that way. It is different with every singer; they each develop their own technique and sound which works for them.
I'm not sure this is really an answer, but probably I'll ramble on for way too long to be a comment.
Tremelo/vibrato can be stylistic or masking; blending or highlighting.
Genre will also dictate which of these may be 'compulsory' for that style.
Opera, for instance, appears to make it almost mandatory [not a genre I'm a fan of]
I've heard many instances where the trem/vib is used to hide the fact the singer really would not otherwise be able to hold the pitch. [citations if necessary, but I don't want to point the finger]
Once you move to more modern styles - anything 20th century or newer - the stylistic aspect softens, but may still be 'compulsory' for that style.
Think Bing Crosby [still not my favourite genre but the guy could sing!]
There is a gentle increase over longer notes, softening & blurring the perceived pitch.
Using this style of vocal in a choral or backing vocal aspect will broaden the perceived 'size' of the 'choir'. The human ear likes broad, even an untrained ear.
A trained ear, or someone [me] with very little tolerance for tuning inaccuracies can hear the 'centre' of the sought-after note without cringing or leaving the room.
In the past 20 [maybe even 40] years, zero-trem/vib has been more common, for untrained singers who simply never learned to do it, never thought it necessary, or simply use it more judiciously. [Example: Tom York, Radiohead - seems to choose not to use vib, but can when he wants, to good effect.]
Hitting a big 'money note' with no trem/vib takes nerve, talent & very serious breath & pitch control.
Edit
I just thought of Oasis 'Wonderwall'. The vocalist has no vib, no real pitching - yet unless you actually pitch the 'money notes' in the chorus sharp [like he does], it sounds 'wrong'. I've never figured out why, it might be something to do with the uncertainty of 3rds.
On your last point - I'm not sure that, other than for some very specific short-term effect, I would ever try to synchronise vib/trem to the song tempo.
Vibrato should usually be a natural phenomenon of a relaxed powerful voice but with the advent of amplification it has changed to more of a stylistic choice.
Check out Nina Hagen's "Naturträne" and compare it with other songs from the same punk album (like "Unbeschreiblich weiblich"). It is obvious that her use of her operatically trained voice is both deliberate as well as organic. Of course, use of the screeching unmodulated voice is also deliberate.
If you had to guess which of those two performances left her more hoarse afterwards, and which one would have been intelligible from a larger distance without amplification, you might arrive at different outcomes.
I can briefly explain what vocal vibrato means to me; it means that you are currently executing phonation with a perfect mouth shape and inner-space.. hence a natural vibrato is the result (it's not actually "forced"; but allowed to happen naturally).
The first time I did it I was blown away.. because I didn't "do it". I practiced proper technique and one day it just started bouncing up and down.
When to do it? That's depends on the style. In general I will say that a long note naturally tends to "want" a slight vibrato as it tails off. Sounds in nature tend to do this as well; so perhaps it's ingrained in our hearing.
Vibrato tends to be around 8 cycles per second; give or take depending on air flow. To sync with an underlying rhythm would sound odd (to me) but possible if you "fake" it and sing it that way.
The singer of AC DC should not try vibrato. Sounds silly when a monotone would fit that type of music. Several other artist would fit the same category. When it sounds contrived it's not good
In rock and pop vibrato is undesirable. Someone who can sing "straight" is more effective than someone who wavers, intentional or not.
In modern rock and pop, someone who uses vibrato cannot be autotuned and therefore is considered useless.
I wish this wasn't the case, I'm a big fan of depeche mode - both singers use heavy vibrato. But it's been over 20 years since they were relevant.
Freddie Mercury used vibrato. AND he can sing perfectly straight. A true master.
Listen to Blink-182 for a minute and then the thousands of pop punk singers who copied his accent / style. There is an anti-vibrato culture
