Lee's strategy was to engage the enemy and destroy his army in detail.
He fought at Gettysburg because that is where his forces found the Union forces.
If he defeated Meade at Gettysburg and destroyed his army, the only thing standing between the ANV and Washington DC would have been the understrength perimeter troops.
Lee was known to suffer from angina pectoris, and has been said to have been having pains during the battle of Gettysburg. Some historians believe that his angina may have affected his judgement at Gettsyburg. One theory, anyway.
Lee did not destroy the Union army at Second Bull Run, or Fredericksburg, or during the Seven Days battles, or at Antietam or Chancellorsville. What makes you think he would have destroyed it in Pennsylvania?
Not quite, Washington DC was one of the most heavily defended cities on the planet at the time. It was ringed with over 50 forts, with 800 cannon, with 20 miles of rifle trenches and hundreds of battery emplacements. Nearly 40,000 men manned these works. this large concentration of troops in Capitol meant that these men were not available to reinforce Army of the Potomac as it marched north. General Hooker resigned as the AOPs commander because Washington would not release any of the garrison to the AOP.
And failure to keep Lee out of Washington DC would have meant murder of government officials, burning of government buildings, and destruction of government records.
They didn’t want to ‘just be left along’. They wanted the whole US to be subject to the slave power.