I was moved in 1996, new job took care of all the details, the movers were fine, it all worked. In 1996.
Today I'm a trucker. Right now practically all shipping companies are overbooked. My company is paying us a "productivity" bonus for increasing number of miles driven. Everyone is buying stuff, and everyone is moving. So it wouldn't surprise me if your stuff gets there late, because unfortunately your shipper (or the broker) doesn't want to leave money on the table merely because they don't have the trucks and drivers. As ugly as that is.
Right now there are many trips that are late for delivery on the day I pick them up. My trips are typically 500 to 1500 miles and take one to three days.
From other threads, do it yourself seems like a good idea. Or maybe just DIY for some of it, and have the rest moved. You may have trouble finding a rental truck exactly when you need it. Because everyone is moving.
The "load it yourself and we'll pick it up and deliver it" solution is good. You can load the container at your leisure. A friend did that, worked out well. Again, around 1996. These also probably have availability issues right now.