Shaking

Normally, I'd tell you to make sure the nuts and bolts securing the rack and pinion are properly tightened. Then, use a medium-sized pair of slipjoint pliers to squeeze the tie rod ends. If they move more than 1/8 of an inch, replace them. Them, turn the wheel all the way to one side and squeeze the boot on the steering rack until you can feel the inner tie rod socket. Then have someone try and turn the wheel (not the steering wheel) by pushing and pulling on the front and back of the wheel while you squeeze. You should feel no relative motion between the socket and the tie rod arm. Next, raise the front of the vehicle and use a prybay to try and move the wheel up and down while observing the ball joint. Repeat on both sides. Make sure the struts are properly torqued, top and bottom. Then, make sure the CV boots are in one piece and not ripped. With the front wheels off the ground, rock the tires at the top and bottom to check for play in the hub bearings.

But that "Hot months only" part is weird.

Driving actually heats the tires up much more than outside weather conditions will. You can drive on an icy road and the tires will heat up almost as much as if the road weren't icy.

Have the cooling system checked, also. I don't think it has any bearing on the issue, but that hot months only still has me confoozled. See if you can get a scan tool hooked up to the car and read the codes. Try and do this when the problem is occurring. It MIGHT be something with the transaxle, though I doubt it.

Since problem started after tires replacement and seems more evident at front left, try rotating the tires to see if thr problem migrates.

Are the rims OEM?

Are the rims OEM?

When tires were rotated, were the symptoms exactly the same or were there any difference in the shaking and source?

Since the problem hasn't changed with a tire rotation or alignment or wheel balancing, check the suspension components listed in my post to make sure they're properly torqued and not showing excessive wear.

Yes, I agree with checking the suspension components and drivetrains, espcially the axle inner joints.