Sounds like a pretty honest lawn company, which is more than you can say for
most of them.

If you believe that lawn chemicals end up somewhere you don't want them
(which is true, but many people like to pretend they don't know), you can
still fix the lawn without them. But, there's one factor you didn't mention.
How much patience do you have? If it took two years get grass to dominate
the weeds, would that be acceptable? This is possible with nothing but
watering, seeding and mowing properly. As far as it being compacted, you can
hire a landscaper to do that.