Origin of the lavalier microphone

According to Wikipedia, the lavalier microphone made its debut around 1932, and it was used in television and studio applications. The main thing to note of this initial use case is that television studios do not usually have "sound reinforcement" (live) systems amplifying the speech in the same room as the person wearing the microphone. There is no concern with feedback, because there are no loudspeakers amplifying the voice of the person speaking. However in the 1950s, the lav mic started to make it's way into live sound applications, such as studio audiences, where the person speaking not only needed to be heard for recording and broadcast, but also amplified so that the live audience could also hear them. This naturally made it into the church, and its usage grew significantly. 
 
In the 1960s, PA systems like the Shure Vocal Master column arrays did a decent job of reinforcing the speech in the room because their 100w amplifiers being more than adequate to amplifier the 2-4 channels on their mixing console. In many cases these early systems were simply "reinforcing" the acoustic speech coming from the pulpit- the lavalier mic could be placed on the pastor or minister inconspicuously, and feedback was not a huge obstacle due to the lower amplification factor of these early units (at least compared to today.)

Changes in live sound

Between then and now, the general public has experienced a monumental increase in the quality of audio experience (and for that matter, video, with one of the secrets to good video being good audio.) AM radio went to FM, movie theaters employed multi-kilowatt sound systems, and our cars have been turned into high-fidelity mini concert halls. This has (like it or not) raised the bar as to what church congregants walk in the door expecting to hear (the debate about the church racing to meet this expectation is a topic for another conversation!) Speech reinforcement has transitioned into the PA being the primary source of audio in the room, and those multi-kilowatt** sound systems from theaters and concert venues can now be seen at the corner church - which now has a full worship band, in-ear musician monitors, and a 24 (or more) channel digital mixing console.
In among all of this change, the poor-old lavalier microphone has been called upon to do more than it was ever designed to -  even in light of the much-higher gain amplifiers being used. Those early broadcast-only lavs were omnidirectional (some of the best sounding IMHO.) But as the PA transitioned into being the high-gain primary source of audio in the room, feedback continued to be a problem, as well as the mics picking up stage noise. Cardioid lavs came out, then super-cardioid- and even hyper cardioid can be found now. And with that directionality, there's a struggle to get them to sound like a natural voice, as they tend to pickup from the throat instead of the mouth.

The struggle is real (with lav microphones)

Ask anyone at your church running sound, and they will share with you that the lavalier mic is one of the largest battles that a modern church sound engineer struggles with.
 
Meanwhile we now have available to us high quality, nearly invisible headworn condenser microphones, that place the microphone capsure less than an inch away from the source of the sound—the mouth. The sound difference between these microphones and lavs is night and day. The end result is a clear message for the congregation, and nearly perfect capture for online streaming and way less struggle for your sound engineer.

What is a church to do?

Sound systems today have grown to a point where lav mics no longer work well—just ask your sound engineer. The clarity is not there, the struggle with feedback often prevents the voice from being set at a proper level for all in the venue to hear well and clear. Just to clarify, I'm not advocating the use of headworn microphones. I'm simply an advocate of what works and lav mics simply don't do the job at the level the listener is expecting. Todays churches with the best sounding speech audio are using either headworn microphones, or handheld mics. Some are using podium microphones, but these can also have issues similar to lavs that are difficult to address in todays modern church sound system.
 

Why is your church still using lavs?

The main reasons that I hear for this is, the leadership either doesn't like the look or feel of headworn mics, or there is some concern over them presenting too "production" oriented of an image. As someone who has served the House of Worship segment of professional audio for 20+ years, I want to point something out (with all the love and respect that I can offer in this printed media):

If you are still using lav mics in your church, there's something more important to you than the clarity of the message.
No harm meant by the above statement. It's simply physics. With today's high-gain PA systems, the lav mics placement is too far away from the voice source, requiring it to be "gained up" to a point where it also picks up the same voice from the loudspeaker, starting the feedback loop. Head worn and handheld microphones can he held close to the mouth, requiring less gain, and they are less likely to "hear" the PA speakers, making it harder for them to start the feedback loop.
A lav mic will never do what a headworn or handheld mic will do for your speech audio quality.
And this may be totally okay for your intended purpose. I just beg you to please be honest with your intentions. If you don't like the look of head worn mics, and you are struggling with sound quality from your lav mic, then visual perception is likely more important to your organization than clarity (and I am in no way here to judge this). Just know that if your visual concerns won't allow you to choose a better, more clear sounding (and less difficult to use) head worn or handheld microphone, then sound quality is taking a subordinate position to visual in your organization.
 
My point is in no way to be harsh, just know that there are better solutions than lavalier microphones today, if sonic clarity and ease of use are on the top of your list. If they are not, that's okay. Just know that speech quality will suffer somewhat and make sure that the way you approach this is in line with your organizations values.
 
"But such and such pastor online uses a lav mic", you might say. And while there may be a few left that do, I can tell you that if you analyze the tech budget of that organization, and compare it to yours, you just might find that they have a bit more resources than you. Can a lav be made to work? in some cases, yes, however to get it right, it may require additional expense, and possibly be beyond the skillset of your volunteer sound engineer.