When reading about the Napoleonic Wars, one thing that stands out is the fact that Britain was Napoleon's main antagonist throughout the successive wars. However, direct engagements between French and British troops were much rarer than, for example, confrontations with Austria. The Napoleonic wars can be seen as a Britain keeping France at arms-length until France's resources had been depleted.
Since Britain is much closer to France than the direct participatans of the war, one would expect frequent clashes either in the English Channel or via expeditionary forces. Napoleon dreamed of invading Britain to deal with "the nation of shopkeepers" once and for all. The obvious reason this did not happen is that miles of coast are much easier to defend than conquer (Omaha Beach comes to mind). After the Royal Navy sunk the main parts of the French fleet at Trafalgar, an invasion of the British Isles was de-facto impossible. Any invasion would have had to rely on favorable weather and hope to not be detected by the superior Navy ships.
After the French fleet was dealt with, Britain blockaded French ports in and around the Mediterranean and seized French vessels. Britannia ruled the seas and had a world-spanning empire. Britain was out of Napoleon's grasp.
Why did Britain not invade France (earlier) and put an end to French wars on the continent?
Britain was a naval power akin to the United States in the 20th century. Her wealth and dominance came from a world-spanning trade system that allowed for the global flow of goods to and from Britain (denoted in GBP) and an industrial base that was technologically superior. The first Industrial Revolution meant that British productivity1 and industry outpaced the rest of the continent (apart from the Netherlands)2. This meant Britain had a lot of capital.
It could have raised a large continental army and fight on the continent itself, but at great risk. It would have been a huge logistical undertaking to supply an army across the water in enemy territory. The clashes between continental forces in the Napoleonic Wars resulted in the biggest loss of human life at least since the Thirty Years' War in Europe3, making deploying a full continental army unattractive. Britain deployed troops in Portugal during the Peninsular War, landed an expeditionary force to retake Holland (which failed), and aided the final coalition as Napoleon was already severely weakened after losing in Germany4.
Britain indeed had experience with continental armies in the form of the troops used by the East India Company during their conquest of India. Crucially the EIC was using primarily Indian troops (sepoys) led by British leaders5, rather than troops comprised of a majority of British soldiers.
What if Britain could have others fight for her? It used it's dominance of commerce and the sea to prop up France's enemies on the continent and have them fight the French armies and threaten the French homeland. It outright paid Russia, Prussia, and Austria to fight France on multiple occasions. It used it's diplomatic and economic ties to bolster the enemies of France. It supplied weapons and supplies to various allies all over the continent and supported insurrections against French rule6. This strategy resembled modern Lend-Lease strategies. All of this meant that Britain had no desire to raise a large army herself but rather have her allies assume the human cost of the fight against Napoleon and the destruction to their territories.
The fact that the British Isles were never under direct attack meant that they could keep expanding their trade networks, grow their economy, and keep supporting its allies.
Napoleon saw some of this and tried to weaken the British economy through a continental tariff regime but never fully succeeded. One could argue that a massive economic downturn might have ended Britain's policy, and led to a collapse of its support for the various coalitions, but the higher productivity and her colonies meant she had ample resources. Britain spent a huge amount of money on supporting her allies, but it was a cheap bargain compared to the cost of fighting herself.
This in the end meant that Britain was the main winner of the Napoleonic Wars. France suffered a humiliating defeat, but even the "winners" like Austria, Prussia and Russia paid a huge price. They lost a substantial portion of their working age population and suffered destruction of argriculture and cities. Britain was the sole super-power. It has spend a lot of money on supporting it's allies, but it avoided the cost of destruction at home. The strategic support enabled by the empire and technology was a bargain.
https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=8c94b8f3de290cb3190f5704df516081e197facc ↩
https://econ.columbia.edu/wp-content/uploads/sites/18/2021/11/bns_malthus.pdf ↩
https://www.napoleon.org/en/history-of-the-two-empires/articles/napoleon-the-dark-side-the-human-cost-of-the-napoleonic-wars-3-min-read/ ↩
https://www.rusi.org/explore-our-research/publications/commentary/coalition-diplomacy-french-revolutionary-and-napoleonic-wars-great-leap-forward ↩
https://www.worldhistory.org/article/2080/the-armies-of-the-east-india-company/ ↩
https://en.wikipedia.org/wiki/United_Kingdom_in_the_Napoleonic_Wars#:~:text=Through%20its%20command%20of%20the,Peninsular%20War%2C%20Britain%20played%20a ↩