I have seen power companies use a recording device that hooks up to your line and records on a graph what happens. If left on for 24 to 48 hours it should show what is taking place. To just arbitrarily replace the transformer and reconnect all the connections seems like a poor replacement for troubleshooting the cause of the problem. They may or may not be lucky enough to eliminate the source of the problem with that course of action. It is like replacing the battery, then the alternator, then the switches in a car to get the headlights to work. You get a lot of new parts and it my cure the problem, but it may not.
If you have any friends with an infra-red camera or visual device you could look at all the connections on your side and see if there are any hot spots. Loose connections are always a source of heat.
The problem you described could be a number of things so you have to keep an open mind when troubleshooting. The thing that stands out is the yard light that is connected on the line side of the meter. If I was a lineman troubleshooting this, that would be a red flag. But that does not mean it is not something other than the transformer.
If you have had recent storms in your area, the power company is likely super busy with repairs and brushed you off as a low priority. You did good to go talk to someone higher up.