Thank you for your response. I have been doing a brand new clean install each and every time. Have formatted the card between installs (I know this should not be necessary since you are then just going to write over the card with an image), and have used two different micro SD cards for each of the installs, one 16GB, the other 32GB. Both cards are SanDisk and both are new. Thus, I do not believe the card(s) (or any of the hardware) is at issue.
As far as I know, I am using the official site to download the LibreElec install file, https://libreelec.tv/downloads/raspberry/. I have located the page showing mirror sites and would be glad to try downloading from the Netherlands, for example, or some other non-US location. However, I now believe this also would not address the issue because...
Based on your comment of the log file indicating the temp file cannot be written to, I went to System Information / Storage, and I see the following:
Filesystem Size Used Available Use% Mounted on
/dev/mmcblk0p1 511.7M 141.9M 369.9M 28% /flash
/dev/mmcblk0p2 26.0M 21.4M 3.9M 85% /storage
Given /storage is 85% full, I assume this is why the temp folder cannot be written to. The installation of each skin / service / add-on must be requiring more free space than is available for all the files each pulls for installation. So, the question now becomes, how / why is this happening? It makes no sense to me that the sizes shown above are in MB. The card used this time is the 16 GB card. It was formatted to 16 GB. The Raspberry imager sees it as 16 GB when the LibreELEC-RPi4.arm-10.0.1.img.gz image file is fed to it. Something is causing these partitions to be set too small when the LibreElec image is written to the card. I do not know how to resolve this.
For comparison, on the 32GB card, I presently have loaded LibreElec 9.2.8 / Kodi 18. It shows:
Filesystem Size Used Available Use% Mounted on
/dev/mmcblk0p1 511.7M 144.4M 367.3M 28% /flash
/dev/mmcblk0p2 28.3G 168.8M 28.1G 1% /storage
Not apples to apples obviously, but the point of this comparison is that /storage under LibreElec 9.2.8 is using the vast majority of the card's overall space, not a very small portion as I am seeing under LibreElec 10.0.1. This appears to be the issue however I have no idea how to troubleshoot / resolve this. Found some instructions via Google that provided some detail on how to resize partitions but upon attempting, I receive a bunch of messages that such is not allowed under LibreElec for security and stability reasons.
Is there a way to manually force partition sizes during the LibreElec image burn to the card? The only override, if you want to call it that, that I know of is CTRL_SHIFT X which I have used to pervide network credentials, enable SSH, and set time zone as a part of the image burn.
Again, thank you very much you for insight here.