I have had this same problem at home as well. I just got my model 3 last week. I am also in the middle of troubleshooting it like you mentioned, and ordered another wireless node to put near my garage thinking that was the issue (just like you).
One of the strange things is, my base home router is a tri band one that has (1) 2.4 ghz and (2) 5ghz networks. The networks have different SSIDs so I have network XXXXX, Network XXXX-5g and network XXXXX 5g-2. The tesla in the garage sees the 2.4 and 5g-2 network but does not see the 5g one even though all three are broadcasting from the same device.
I tried connecting the tesla to the XXXX 5g-2 network (because the 5g ones are faster but have less range) but same problem. I also tried connecting to the XXXXX network (which is the 2.4ghz one) thinking that the tesla may have an issue with the 5g band even though it appeared to connect successfully.
I have been letting the car go to sleep for at least an hour or so without accessing the tesla app, and I dont have any other third party tesla statistical apps running.
The last time the tesla app could not "wake up" the car, I went out to it, and it unlocked fine. when the screen woke up, I could see that the LTE icon was disconnected (had a slash through it). No wifi connection and LTE down. I tapped the LTE icon and checked the wireless networks, there was still a checkmark next to the network I had the car on. during that time LTE connected.
So, thats why I thought it might be a range issue. Although I have a pretty high end router for home, its upstairs and away from the garage so I thought I needed an extender nearer the garage. I just ordered it yesterday and it hasnt come from amazon yet.
It appeared to me that the car goes to a deep sleep and shuts down the wireless during that period of time. My car is getting tinted at a local tint shop as I type this, and I have checked the app several times with no issues on connecting to it (and seeing various doors open on the graphic in the app, which is cool as @#Q!@# to me lmao).