Which vendor are you using for Travel Management?  And are they also your expense management vendor?  Lastly, would you recommend the solution to others?

We currently use Concur for processing all T&E expenses. We have our corporate T&E credit card program connected to Concur and then another connection to NetSuite. The process, including receipts and approvals are digital and automated. It has been working well for us. I recommend it.
We are not using the Travel Management feature of Concur just yet.

We work with American Express. They manage the travel reservations including hotel and transportation, including car rental. However the only bill airfares directly (as I understand through my own experience) while all other expenses are managed internally. The service in scope works well and you raise an interesting question as I was not aware the travel vendors could manage the entire process. Let me know if you want to deep dive on this.

We currently use Concur for Travel Management reimbursement. We book our own travel.  Concur works well.  However, since we are implemented Oracle effective July 1, we chose a different solution.  Please feel free to reach out to me for more information.

We do not use a travel management company; our travel services are done in house.  I'm not sure the cost benefits of outsourcing but I'm sure its not a difficult cost benefit analysis to do.