Sounds like targeting "city + service" with the home page is best in your situation. Don't make a separate local page for the city for the reasons you've mentioned (plus your home page will have more ranking power).
Instead create local landing pages to target regions (if applicable) or...
It's definitely a linkbuilding scam, I've seen the same templates. Big G has been alerted to it and John Mueller was asking for examples a little while back. More info here:
https://www.searchenginejournal.com/googles-john-mueller-to-investigate-deceitful-link-building-practices/350066/
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