I have twice missed my connecting flights and had to make my own flight arrangements, once flying to Warsaw right after 9/11 and one flying from Miami to Guatemala City.
In Warsaw, there was a GCT person, but he wasn't there to meet me. He was there to meet six people from San Francisco. I had called the GCT emergency number to give them my new flight, but they never passed it on. However, we found each other and he took me to the hotel. I was really nervous about taking a taxi on my own with the language difficulty. But if would have, if it were necessary.
In Guatemala City, there was also someone there to meet me. And there would not have been such a language problem.
And I don't think it would have occured to me either to ask for reimbursement if I'd had to take taxis.
When I flew to Shannon a day early, GCT made my flight reservation at my request. I wasn't charged extra. Made my own hotel reservation and got myself back to the terminal to meet with the group for the trip to Galway. The driver had my name, so that worked out fine also.