For business, our corporate TA is great and also has 24/7 backup which, among other things, proactively monitors and handles reroutes when a misconnect is likely. I have found the TA good at securing overbooking when I need a flight and in locating hotel rooms when there are none.

On the misconnect situation, I'm not stuck in line. I have an email & voicemail when I land with the details of what's next. Sometimes it's just a text message such as "run to Gate 15". If there is an overnight, there is a reservation for me along with directions, so I'm not stuck waiting behind 180 people.

For leisure, I do my own except for complex vacations where I use a TA who has experience with the specific type of trip and who comes recommended by friends I trust. It's more than worth the fee to have someone who has a relationship.