My mechanic won't pick up
Solution 1:
Check if you have any cars to have your mechanic deliver to you. If you have one car and it's seized or destroyed , he won't pickup as there are no cars to deliver.
Try calling Insurance instead.
Solution 2:
An important detail is that the mechanic will not be back in service until you have paid all the mechanic and utility fees. Have you ensured that you have?
If you haven't, you need to freeroam until you get a message saying that those fees have been paid. The game will prompt you when that happens.