Anonymous comment from a recovering alcoholic:
When I’m late to church, people turn around and stare at me with frowns of disapproval. I get the clear message that I’m not as responsible as they are. When I’m late to AA, the meeting comes to a halt and everyone jumps up to hug and welcome me. They realize that my lateness may be a sign that I almost didn’t make it. When I show up, it proves that my desperate need for them won out over my desperate need for alcohol.
My Spanish comprehension is sketchy, but I remember a visiting priest starting off a “homily” with a few minutes of criticism about the Mass starting late, and people still streaming in during the readings.
When one thinks of Mass as less an obligation and more an aspiration to saving grace, it casts lateness–even for mundane reasons–in a different light.
Catholic Sensibility 
That sounds like one unusually prompt church; I can’t help but wonder if it’s Catholic or not, because an elastic relationship with time appears to be so common at virtually every Catholic parish I’ve ever attended that it’s hardly noticeable.
PS: when the celebrating priest is chronically late*, however, it’s often taken at best as an invitation for everyone else to get more elastic in terms of time, so if-we-wait-for-everyone-before-we-start is never a successful response in the long run.
* Said priest making a joke of it is disrespectful. Chronically joking about an annoying character trait or habit to those who experience the trait or habit say “I know this is [insert negative adjective here] to you, but I am in charge and there’s nothing you can do about it or me.”