haxchi's purpose is to stop using browser hax completely.
Haxchi IS a homebrew enabler and launcher.*
additionally, it also became a firmware's signature patcher, but its first purpose was a way to access homebrew without the need of online connection, because the browserhax is unreliable and freeze too often.
just use your haxchi channel with the corresponding button to launch Homebrew_launcher.elf
once in the homebrew launcher, launch haxchi installer again.
haxchi and cbhc on your console are nothing more than an already replaced file. overwriting it again will only re-replace it.
you use a NDS retail game emulator, which contains a ROM.zip (official NDS game's ROM). When you use haxchi installer or cbhc installer, it replaces that file with haxchi's ROM.zip
if you use haxchi's installer, it also copy the config.txt along the rom.zip file.
if you use cbhc, it edits the boot config .xml file used by the console to auto-launch (coldboot) that haxchi ROM.zip at console boot. (cold boot hax chi = CBHC)
*
As a homebrew enabler, it's (1) patching some part of the console's memory to allow user level homebrew launching.
Then, it (2) launches the homebrew set in the config.txt (usually homebrew launcher by default).
If you try to re-launch HBL using haxchi's channel default behavior, it will not only tries to launche HBL (step2), but also will re-attempt to do step 1: patching the memory to allow homebrew launching.
Doing step1 again might freeze the console. This was a problem with homebrew browser, and to fix this issue the browserhax did something different :
(1) patch the firmware to allow user level homebrew launcher
(2) patch miimaker channel, to add homebrew launcher.elf into that channel (not replacing it, just adding it FIRST, if you exit hbl it launches miimaker)
(3) launches miimaker channel, which launches HBL.
if you want to launch HBL again, don't do browserhax again (it will freeze at step1! ) but launch MiiMaker channel, which is the one launching HBL.
I'm not sure haxchi has a similar fallback like that to prevent step1 repatching. Some users reported freeze upon relaunching haxchi's hbl forwarder.
I never used it, so I can't confirm myself. I should try one day.