Do most U.S. cities, even small ones, now have personnel policy
manuals which detail the recruitment and selection processes for all
employees, including city managers?

My guess is 90% of cities have at least some guidelines written down
by someone as a guide in selecting new hires. The policies may be
unofficial and/or rarely followed even if they have been approved by
the city council. I believe less than 1/2 of USA cities have a city
manager. Typically a separate manual is used for the city manager and
other top positions.  Neil