Are Nice Condition LMC's Worth Anything?

Generally speaking yes. As said above they need to be say MS67 or higher to be really worth more,
of course depending on year, MM etc. IMHO

That is a long and wordy description to ask a question ... but what is the question?

Some can be of very high value, some not so much. What is recommended is to look at the site John1 mentioned, or also can go to the graded prices on sites like PCGS and NGC. This way you can see the value if a coin was sent in for grading and came back in that specific grade. Modern coins generally its the top condition or only the top few that have value, while the rest tend to be under the value of the grading fees so you just need a bit more caution with moderns. You can also check out the census on PCGS and NGC, meaning they also list the number graded in each condition. If only say the top grade has value worth grading, yet only 1-3 out of millions have been graded in that condition, its far less likely that you have another and need to develop some skills to help find those high grade coins. The best situation for a person newer to grading is when many of the graded values besides the very top condition which is usually rarer, are over the cost of the grading, therefore its not a potential loss for misidentifying a top grade at least. But back to the question, yes if you look some do indeed have high values in top grades and should not simply dismiss looking when you potentially have one.

Repeat, unless you know your stuff, I'd cash these in.

Each coin has to be judged on it's own merit. For me, "dumping" the coins into circulation without evaluation would be crazy. Memorial cents can hit 4 figure+ values.

Prices are low due to lack of demand. So long as there is almost no demand prices will remain low. Some specific dates are in such high supply it's unlikely there will ever be enough demand to affect them very much.