Shouldn't cities be obligated to make sure they're able to have teachers and plumbers and piano tuners?
There are alternatives for everything with their own price. Even if it is horrific like "die of rabies instead of getting vaccinated after being bit".
If availability of workforce in a broader area is high but in city housing they afford is low either they need to be paid more or traffic and commute time will increase and make the job less desirable which may affect skill level and harm labor availability. Underserving is technically also an option but there will be knock on effects - if the schools are bad then parents will be more likely to relocate.
Not all parents. Only the intelligent, cultured, and high income ones.
If rent gets so expensive that plumbers can't afford to live there, then some plumbers will just start coming in from some cheaper area and charge customers probably an increased fee based on the trouble of getting into the city.