Absolutely embrace the warm weather zoysia. Nashville is in the transition zone, meaning both northern and southern grasses can thrive there; however, with the Earth warming, you should start to favor the southern grasses. Fescue does not spread, at all. Zoysia will, as you've seen. If you want the zoysia to spread faster, take plugs from your own lawn to ensure you have exactly the same variety of grass.

Speedwell is a problem for fescue, but should not be a problem for the zoysia. Mow the zoysia at the mower's highest setting. If the speedwell is not choked out by the density of the zoysia, you might need to spray something.

Generally, if your front yard is not looking great, that indicates a watering problem. You should be watering deeply and infrequently. Deeply means 1 inch all at one time. Measure an inch with your watch. Put out some tuna or cat food cans and turn on the sprinkler(s). Time how long it takes to fill all the cans. That is 1 inch of water. Use that time every time you water. Infrequently means don't water every day, or even every week for most of the year. As a general rule, deep water once a month when the temps are below 70 degrees F. With temps between 70 and 80, water once every 3 weeks. With temps between 80 and 90, water every other week. With temps between 90 and 100, water once a week. With temps from 100 to 105, water once every 5 days. You don't have to worry about higher temps, but the point is deep watering and not all the time.