United does not need to reimburse you for this - though if the misconnect was mx they would have needed to provide you a hotel for the night in DEN. Your other option would have been to declare a trip in vain and request a refund as they were unable to get you to your destination for the purpose of your trip, though I have no experience with that.

Its unfortunate that none of the carriers UA does work with were able to get you to your destination - I've run into the same problem before. Just how it goes with who the partners are and are not.

I don't think trip insurance will cover you buying your own 1-way ticket for a reroute, but I haven't memorized the T&C of the trip insurance provider. I think it mostly covers cancellation of a trip due to health or job loss, covering lost baggage replacement, etc.