This should work on a lot of launchers. I will only describe the method fo TL-MCL, as it’s easy to use. Steps everywhere are almost the same (except for launchers who provide things out of the box as is).

Guide for TL-MCL (downloadable from https://ely.by/load)

Create a new folder with TL-MCL, perform a clean install.
Change directory from the default one to wherever your TL-MCL folder is (or keep it if you like mess).
Enable Beta versions (2010-2011)
Select Beta 1.7.3 and open it at least once (so base things get downloaded)
Close the game after that and head over to a folder called versions, inside you should find a folder called b1.7.3 – that’s our instance.
You will find b1.7.3.jar – make a copy just in case something breaks!
Once you’re done, open it with 7zip or any other archive explorer.
Delete META-INF.

Now we’ll be dividing paths:
If you wish to use Mojang’s skin system put the files from SkinFix\Mojang (except for README). If you want to use Ely.by, then put inside the files from Ely.by folder.

I don’t know how exactly Ely.by works, but it’s probably compatible with official Mojang skins (can’t verify).

That should solve the skins issue.

In order to fix sounds – delete resources folder from the game folder and put the one from SoundFix. If you’re on Windows, you may want to right click on it and set a checkmark ✔️ for read only on the folder.

That’s it for vanilla experience.

Launch the game, load a world and see if the skin works (it should work as long as you have an internet connection).
If you can’t hear anything then press F3 + S to reload the sounds. That’s something you’ll need to do every time you get into the game (sorry, no workaround for this I guess).
Anyway, try crafting a door and a pressure plate. These are two critical things that’ll determine whether it worked or not.

Clouds are screwed up – Just suffering from AMD.
All versions prior to Beta 1.8 suffer from weirdly shaped clouds on newer AMD devices (2010 onwards?), in order to fix it, you need to put the content of BitDepthFix inside the b1.7.3.jar. That should work without any hiccups.

Now that’s everything for vanilla experience.

Optional mods:
ReiMinimap – A minimap, radar.
HowManyItems – TooManyItems with a recipe book.
Modern Chat Mod – It’s supposed to allow you to scroll through the chat.
ModLoader – Foundation of mods. Install it if a mod doesn’t work.
OptiFine – Performance booster, good if you’re still lagging. Note that OptiFine for 1.7.3 was still incomplete, thus it may feature some minor or major bugs. The only notable one I’ve encountered was that if you went beyond 10k blocks, snow textures would start flickering.
Xray – Certified hood classic.

Installation of the mods is simple, as you just need to put them inside b1.7.3.jar.
I don’t guarantee game stability with these mods though. It did crash for me quite a bit. Depends on the combination, although OptiFine all alone should do fine.

Download fixes (40MB) - https://files.catbox.moe/kd7nll.7z