Is this article by Shirlee Emmons Vibrato, Wobble, Tremolo, and Bleat a help?
Also see a discussion here on straight tone.
As to your specific question "do people who sing like that know they sing like that" my working assumption is that people sing the way they think they should, more or less following their understanding of what the conductor has asked the choir to do, overlaid by their opinion of what they think the choir needs. They may have a self-image of being a leader. The only way to answer that is to ask them, but I don't think that gets you or them anywhere. IMO the only way to actively work to fix it is for someone experienced in vocal technique to listen and propose changes to fix it. I'm an amateur conductor so don't have in-depth experience to call upon. That's one of the reasons I have asked an experienced chant leader to take us in an afternoon workshop next month.

Does the conductor / music director have the kind of relationship that allows asking the singer to come to rehearsal a half hour early to work through 'techniques of choir blend and balance' privately or for 15 minutes before the others arrive?

A colleague calls them "friendly voices" because they wave at you from across the room ;)

The answer to the OP is no, they don’t know, and they probably aren’t working to fix it. They probably had just enough voice lessons years ago to make them dangerous, but not enough to work out their vocal faults. They probably do not listen to recordings of their own singing and try to self diagnose vocal faults.

That wobble seems to be a natural occurrence over time. In aging choirs which young folks don't seem to want to join anymore, it has become the "church" sound.

Francis, they wouldn't dare wobble around you. You probably already have the anathemas printed and just need to fill names in the blanks.

Seriously, though, I have seen a few local church choirs where those wobbly older ladies are all they have in their choirs.

No anathamas from me! I’m just trying to get to heaven and I just want as many souls as possible to go too!

Seriously, though, I have seen a few local church choirs where those wobbly older ladies are all they have in their choirs.

those choirs are closing down left and right.

We are entirely made up of parishioners. No ringers.

@Cmanfro, the article I linked to above by voice teacher Shirlee Emmons has links to her other articles on breath control.

Vibrato is a natural function of a healthy adult voice

100% agree

@paxmelodious, you should be able to continue your ambitions, if that's what you desire, with the help of voice coach. Disclosure: I'm a couple of decades older than you and a vocal coach helped me pass on the second attempt a routine audition in a very ambitious symphony chorus. There are of course many lessons available on line, albeit very short in order to persuade people to sign up for the full course.

Vibrato (the healthy kind) and wobbliness occurs in men's voices, too.

This is unfortunately very common. Having recently earned the distinction of recognized vocologist (PAVA-RV) from the Pan American Vocology Association, I have more insight into vibrato-related issues than I did when I was working as a choral conductor or even when I was teaching vocal pedagogy at the conservatory level.

As others have mentioned, there are several possible causes of wobble: technical, aesthetic, and hormonal to name a few. There are also solutions, including voice habilitation, vocal coaching, and medical interventions such as hormone replacement therapy.

Ironically, many technical solutions are counterintuitive to those whose training has primarily been in vocal performance or choral conducting. For example, asking a wobbly singer to sing with a straighter tone can actually exacerbate dysfunctional muscle tension and restrict airflow, where a better solution might include reducing or eliminating unnecessary tensions, improving muscular strength, flexibility and coordination, and balancing airflow.

Depending on whether the primary issue is medical (e.g. hormonal changes) or functional (weakness, rigidity, or lack of coordination) a singer might need to work with a laryngologist (not a generalist ENT) or a SVS (singing voice specialist) trained in practices such as Vocal Function Exercises (VFE) and laryngeal massage. But that is only if they’re willing to make the effort to address the issue in the first place.

No matter what, remember that singers are often highly sensitive since our instrument is part of our body. It can be especially difficult to accept that what used to work for us is no longer working. When addressing these issues with a singer, one must approach with appropriate care and sensitivity. But do know that vibrato issues are very common, and that there are things you can do to resolve them.

In spite of the resistance I referenced above, it is actually possible to help a singer improve a wobble if that singer is committed to some fairly serious work on the problem. As was ably pointed out by incantu, above, the first step should be a visit to a laryngologist to ascertain that the instrument is in god shape. The General Practitioner can order some tests to check out endocrine problems (hyperthyroidism or hypothyroidism can play havoc with the voice). Then the work begins. Many voice teachers think that most of these kinds of problems are breath coordination related. Although I think that they are right in many cases, there are other places one might need to begin. It is pretty clear that the whole vocal coordination is involved and a thorough revamping of he technique is usually involved. In a general way, the use of too much weight in the tone is a major factor with the wobble. Giving that up is very difficult for many, but it is often the original source of the problem. Anyone attempting this work should know how to go about it and tread carefully on the ego while guiding a singer with this problem. It is not for the director with some choral singing experience and a semester or two of class voice to attempt this project.

No matter what, remember that singers are often highly sensitive since our instrument is part of our body. It can be especially difficult to accept that what used to work for us is no longer working. When addressing these issues with a singer, one must approach with appropriate care and sensitivity.

Amen.
I used to have one choir member who had directed the choir herself years before my arrival. She had a lovely voice but was starting to decline toward the end of my tenure there. She earnestly pulled me aside and made me promise to tell her once she crossed the threshold, as it were, from beautiful to ...unbeautiful singing. She knew exactly what was going on and had the humility and foresight to prepare herself for that day, whenever it comes. Mercifully, I didn't have to have that conversation with her, but I was always so grateful that she had the self-knowledge and humility to be open to "retiring" from public cantor/solo duties.

Wow… noble humility still exists.