Feeling fine on the Persching line
Persching declined to actively campaign and was regarded as being close to Wilson's policies on many issues despite having Republican leanings. However had he decided to campaign he may well have won the convention and carried on the mantle of Teddy Roosevelt who had been expected to be the Republican front runner had he lived.
The 1920 election would have been a closer run thing as Persching was too close to Wilson and the German and Irish American groups that were anti Wilson and hence Cox would not be enthusiastic about Persching. Persching had German ancestry but was actually against an Armistice in 1918. However the feeling for change would have defeated Cox.
In office it is conceivable Persching could have sold the League of Nations being a former serving General rather than an ivory tower idealist like Wilson. The Washington Treaty would have gone ahead. There would have been no corruption and he would probably have recieved a second term being returned on a wave of prosperity.
He would have followed Coolridge's track record on civil rights having commanded African American soldiers and the Klan would have declined as they did in the twenties and probably stood down in 1928 to be followed by Hoover who would have lost in 1933 because of the crash.
It is possible that with American in the League of Nations it might have taken firm action towards Japanese aggression in Manchuria and stopped the drift to war at an early stage