I am a DIYer who regularly buys houses that others won't touch. It is because I am a sucker for punishment. Also I am in Canada and you are in the American south. Take this post as worth every penny you pay for it.
1. If that house is a single-level, I would think nothing of buying it and rewiring. If this is the case, your wire runs will all be via the basement, tacked on to the floorboards and rising up into the bottom foot of the walls on the main floor. You may have a few spots to patch with tricky fishing jobs, but overall, I would DIY that rewire in a heartbeat.
2. Since the house was built in the '60s there is a good chance that the walls are drywall, not lathe and plaster. If they are drywall, teh fixes for punching through the wall to fish a wire will be a quick, easy job.
3. Replacing a panel sounds big and scary, but it isn't. Around here the electrical authority replaces the wire from the street to the house. They will provide you the jumpers to go from the meter to the panel, but won't hook up the panel until it has a stamp on it from an inspector. The process here would go:
a) Run rough wiring wire to all locations (outlets, lights, fans, water heater, etc.)
b) Turn off power. Remove existing panel, and pony panel. Leave the main, but lock it out.
c) Hook up all the new wiring to the panel and to receptacles.
d) Get inspection.
e) Order new service from street to panel.
4. You should remove all of the old electric that is visible just to give future owners peace of mind, but it isn't at all uncommon to see the old wiring abandoned in the basement, attic, and walls of houses after they have been updated.
In a house that age, you should check for galvanized plumbing. If there is any, you may face a bigger task replacing the plumbing than you do replacing the electric.
If insurance refuses coverage due to plumbing/electric ask for the price of a rider to cover it as an interim measure. you may be able to get a 6-month or 1-year rider to give you time to remedy the problem. If your insurance doesn't offer this, ask for a referral to a high risk insurer who will. I've used Mitchell and Whale locally - it cost me $10K/yr to insure a 4,000SF triplex with galvy plumbing and a few other issues. I got things fixed fast enough that I could ditch the policy after a few months. On the upside, you can negotiate the insurance cost into your purchase/sale price and get the seller to take on your loss.