If you're only seeing the crash with homebrew apps, then its possible you aren't launching them with enough RAM. If you're using the Album to open the homebrew menu, you probably will also see the Applet Mode message indicating that you are launching the homebrew menu as an Applet which limits the amount of RAM for homebrew apps. For full RAM usage, you will need to launch the homebrew menu via title override which you do by holding R over any game/title then launching it.
That error code seems to be associated with the SD card and not the games since it triggers for people without modded consoles. I'm not sure if you ran a legitimacy check but check to see if its a fake SD card or not with a tool such as FakeFlashTest. Its also possible that the SD card is dying.
You're looking at the "missing features" the wrong way. From TX's point of view, they only want to use whatever applications comes bundled with SX OS which severely limits your options (notice how they made SX branded versions of installers and save managers?). Atmosphere doesn't come bundled with any homebrew applications because the expectation is for the user to download those themselves because the homebrew community has developed hundreds of apps. They cannot assume what homebrew apps the user will need or find useful so none are pre-packaged with the CFW (except for 2). However, there are people who offer Atmosphere packages with some homebrew apps pre-bundled. One of them is called DeepSea. Note, they are still offering the same CFW which is Atmosphere; it just comes with some homebrew apps that you may or may not find useful.
The HDD support is a misnomer. Atmosphere fully supports HDDs, potentially more than SX OS. It just doesn't launch .XCI files like SX OS does. However you can install .NSP files from HDDs or even move files from them. And the reason Atmosphere doesn't support launching .XCI files is because of the associations with piracy. But installing .NSP files was deemed fine because there are non-piracy uses such as homebrew apps.
There is no such thing as a "sig patch". What actually exists are ES, FS, & Loader patches. ES patches are short for Eticket Signature patches and these allow .NSP files to be installed then launched. FS patches stand for File System patches and these enable installing .XCI files and homebrew .NSP files. Basically these patches allow .NSP files that aren't signed by Nintendo to be launched. Loader patches are a subtype of FS patches because Nintendo moved the signature checks in their OS from FS to Loader so these patches needed to be developed.