Your first question (and thread's title) is : can I use ONLY the browser exploit. and we answered many times : YES, you can.
but it's a little unreliable, and you might have to try it few times until it works and the console doesn't freeze.
To get a more reliable (100% working) entry point to homebrew, you can use haxchi, or cbhc (which is still haxchi but auto-launched at console boot for lazy people).
What is haxchi?
Haxchi is a hack which replace a NDS rom inside a NDS game's virtual console.
You need a retail NDS game to even launch it without doing a previous hack.
if you install haxchi to a non-bought/pirated NDS game, then you'll need the browser hack to launch a CFW to patch signature check function to be able to launch haxchi, which defeat the purpose of haxchi : not using the unreliable browser hack.
console boot -> system menu -> signed browser -> browser entry point hack -> Homebrew
console boot -> system menu -> signed NDS game -> Haxchi entry point hack -> CFW or Homebrew
console boot -> system menu ->
unsigned NDS game
console boot -> system menu -> signed browser -> browser entry point hack -> homebrew -> signature patcher ->
unsigned NDS game -> Haxchi entry point hack -> CFW or Homebrew
What is CBHC?
CBHC is haxchi, but launched at console boot by the console instead of being launched by yourself from system menu.
console boot ->
signed NDS game -> haxchi entry point hack -> CFW -> system menu
console boot ->
unsigned NDS game -> Bricked console ! no more system menu access...
Can you use browser hack only?
If you don't want to buy a (good signed) NDS game:
you can do the browser hack to access the homebrew launcher. if you don't need the homebrew launcher, you don't even need to do the browser hack after a reboot.
that's the point : You DO NOT ALWAYS need to do the browser hack. you don't always need homebrew or CFW !
To play legit games, you don't need browser hack, you don't need haxchi, you don't need CBHC, you don't need CFW !
so, it's not an issue to use the unreliable browser hack once in a while.
you need browser hack only to run homebrew (example: to install a game), or to play a non legit content (game with a DLC you didn't bought, eShop game bought to be used on another console, game from another region).
You don't need browser hack to play game with a legit key.
what is a legit game?
legit and legal are two different things.
here, legit means the console get what it expects and runs it without the need to patch anything.
The console check if a game is for the correct console. game disc are working on ALL consoles, so if you install a dump of a disc, the console will see "consoleID : ALL" and because the data is not altered by anyone, the signature is good and the console launches the game.
If you install a game originating from eShop (a game which doesn't exist as disc version), the "consoleID information" of the game contains the console ID which bought the game.
If the console check the game's data and see another consoleID, it refuses to launch the game.
To launch on your console, the data need to be changed to "consoleID : ALL" or "consoleID : Your current console ID here".
By editing this game's data, it breaks the signature and the console knows you are trying to mess with it. You need the console's "signature check" function to be patched by a homebrew to force the console to accept the game even if the signature fails to verify.
So, multiple cases to you :
Install
install a disc based game : browser hack > wup installer > install the game
install a disc based game from another region : browser hack > wup installer > install the game. (the signature is good, you can install it)
install a eShop based data (game/DLC) : browser hack >
signature patcher homebrew (not haxchi or cbhc) > wup installer
Play
play a disc based game : nothing needed, power the console and just play !
play a disc based game from another region : browser hack >
signature patcher with region free > optionally run Spiik homebrew to patch region string > play the game
play a game with non legit data (dlc, eshop) : browser hack >
signature patcher with region free > optionally run Spiik homebrew to patch region string > play the game
Play a game with mixed free and non free DLC : there's a trick to put the free DLC's data to a specific location to let the console launch the game without signature patch. Power the console and play!
Other
access NAND : browser hack >
Mocha CFW > ftpiiu_everywhere
Is the browser that much unreliable?
It's random, so it could work on the first try, or on 20th try.
the browser hack used by 5.5.1 was also random, but had more success rate than the browser hack used by 5.5.2
There's a way to replace the browser version to use 5.5.1 hack on 5.5.2 firmware, but you'll need signature check to even launch the old browser while being on firmware 5.5.2, defeating your situation of just wanting to use browser hack without haxchi. (you would need a different entry point for homebrew before launching a non signed browser hack).