Or the condenser has seen better days and is not so far off giving out.
Lots of reasons for burned points. Leaving the ignition live should not burn the points unless they were rubbish in the first place - that usually burns out the coil first. Other reasons could be too points gap too small, a poor lobe on the distributor, or even a worn shaft. Might be others; just have to look for the obvious first and the odd-balls later.