How many men do you think would be needed on the military side of things? Since presumably, something has to happen to convince the Americans that further fighting will end poorly for them.
Amherst not only has to beat the Continental Army soundly, and beat it soundly a number of times, Amherst also needs to somewhat pacify and keep somewhat pacified the territories he's already liberated as he moves on to the next target.
The UK's OTL strategy was an attempt to do something like this. The idea was that the southern colonies and particularly South Carolina would be easier to pacify thanks to the (presumed) number of loyalists, a smaller overall population, fears of a slave uprising, and other factors. As you note, that strategy failed due to a lack of troops among other reasons.
The British couldn't control the countryside, couldn't buttress the Loyalists enough so they could control the countryside, couldn't stop the partisan fighting, couldn't garrison/occupy everything they needed to, and couldn't decisively defeat the Continental/militia forces they faced. Some of those are purely military goals, but most are political ones that military force can only help achieve.
Amherst was a painstaking sort of fellow so I can see him employing a methodical approach and more of a combined political-military one than in the OTL one.