Since you related a scenario tied to your question, I'll try to address both the general as well as specific.
Excerpting from this article:
Davening with a minyan is not just a nice thing to do. It is an
obligation upon every male (above the age of thirteen) to daven with a
minyan every single day of his life.11 According to many poskim this
obligation is d’rabbanan12 in nature, and is a very big mitzvah.13 The
Shulchan Hatohar14 says anyone with trace of yiddishkeit should daven
with a minyan. Although the Shulchan Aruch15 says one should “try” to
daven with a minyan, the poskim say this does not mean to say that
davening with a minyan is not an obligation.16
(I've maintained the footnotes for your xref).
So, there is some debate as to whether one is obligated to pray in a minyan at all. But, this article explains Shulchan Aruch means that it is an obligation. See also this M.Y. question and its answers. Igrot Moshe concurs that the minyan is obligatory.
Well, obviously, this assumes that there is a minyan to attend! So, it seems that where possible, and as much as possible, the shul has an obligation as well to try to have a minyan. So, in that sense, the complainer is correct, that if you've already established a minyan on certain day, the shul really should not forgo that obligation to provide that minyan continuously, since people are obligated to daven at a minyan.
Your problem, is a bit more specific, though. My understanding (and correct me if I assume wrongly) is that your shul encounters these conflicts:
- You're not getting a daily minyan, no matter what happens.
- These people probably don't daven at home. So, it's the shul that's getting them to daven altogether.
- It seems that you are getting a consensus of enough people to make a minyan on Mon. and Thurs. as they realize the importance of hearing Torah reading
So, to me, it seems that the higher priority should be given to keeping the shul available to have as many minyanim as possible, and moreso, getting these people to daven altogether, if it's the shul that encourages them to do that. My inference, then, from the general joint "obligation" of a minyan is that you may have to pick and choose on this one and side with when you can have minyan the most. If it means dispensing with the Wed. minyan, you may need to do this.
This is my opinion and inference from the above article. CYLOR. Even better, I recommend that the "blackmailer" ask a rabbi about this, and have HIM post the answer on M.Y. ;-)
I reread your question, and you imply that there is another synagogue in your area. If that's the case, then, it's possible that none of what I said may be relevant. Can you clarify this aspect?