Dreadful! Isn’t Europe supposed to be part of the modern world?
They do that in London too, special shuttles that run between Gatwick, Heathrow and the other airport I've never seen. My last visit I took the 6 AM train from the town where I was staying over to Gatwick, then the hour-long shuttle to Heathrow for my flight out. For FAR less than the $100+ a taxi would have cost.
Other options include trains. From several airports in the area I could possibly take a train into Strasbourg. Or renting a car -- expensive but sometimes necessary.
Sometimes all the options make it hard. There are 5 or 6 airports within 70 miles of Strasbourg!