Although this works for me consistently, I currently have not had anyone else try this successfully so please let me know if you get this working. 

Crysis Quicksave mod
This is a very basic mod that enables Quick saving in Crysis 1 remastered with no other modifications to things like graphics. It currently requires devmode to be active for actual Quicksaving since the save command is locked to devmode only.

Features:

• saving  (does need devmode)
• loading (does not need devmode)
• removal of cheat functions(only for non-cheat version)
  

Limitations:

• Devmode is required for saving, it adds some weird icons whenever it is saving or loading assets, this might be able to be turned off, but I have no idea how.
• Cheats and unrestricted console access comes with devmode and those kinda hurt the gameplay, accidentally teleporting can be a bit un-immersive. 
• Devmode also resets graphics settings each time, requiring you to change it back every time.
• Load and load from last save both load the last save, the menu for save browsing no longer exists.
• The teleport to next checkpoint command is not present in the file I found so is still present even in the no-cheats version :(.
• This will only load the last save but the game internally keeps the last 4 saves and overwrites them with the oldest being overwritten first. Don't go saving in terrible spots otherwise you may not be able to recover. You can go to C:\Users\{YOUR_USERNAME}\Saved Games\CrysisRemastered\SaveGames to see the quicksaves created and back them up or use an older save.
• After testing, animal states seem to not be saved although that might be present in the original as well. Just killed a turtle and he did not return when I reloaded, I'm a monster. Everything else seems to work fine.
  

Keyboard Shortcuts:

• F5 - Quicksave without checkpoint
• F6 - Quicksave with checkpoint
• F8 - Load (does the same thing as load last save)
• F9 - Load last save 
  

How I found it:
I bought Crysis 1 remastered with high hopes and great expectations. I was hoping for the best version of Crysis and the several patches that should've fixed all the problems. The result was maximum disappointment. The new raytraced stuff was cool but a lot of the cutbacks from the console port are still present in the third and final patch. The most important missing feature for me was Quicksave. I hate not controlling when I save and often the games checkpoints occur in awkward and sometimes unrecoverable positions. I decided to dig through the games files and see if I could find some commented out code related to Quicksaving. I pretty much did exactly that.
I extracted all the pak files using 7zip (since they were just zip files in disguise) and searched for words in the files like 'save', 'load' and 'F5' (the key for saving). 

After digging through the extracted files I found the code that enables the key bindings for 'save', 'loadLastSave' and 'load' commented out. After enabling these loading from last save worked (loadLastSave and load do the same thing in remastered) but save still didn't work. The command was clearly not incorrect as changing 'save' to 'checkpoint' would give unknown command errors while 'save' gave no errors, so it was recognised as a valid command but did nothing. I ran the game in devmode, which normally allows all valid commands and save started working!! I even found that 'save_quick' also saved the game but in addition showed the 'CHECKPOINT REACHED' box when pressed.

Going through the games pak files you can see things that might be interesting to some people like:

• Code for gboots (possibly gravity boots) and other things commented out hinting at cut features.
• Saber Interactive fixed the badly indented code in a lot of files and removed some code that seemed to do nothing.
• Mini comments about who made certain changes and comments saying TODO for bugs that have clearly never been addressed lol.
  

This was fun to find but, in the end, neither the original Crysis modded nor Crysis remastered fill the void for me for a definitive version of Crysis. I haven't checked whether this works for the whole game and not just the first 2 levels but I got frustrated by constantly standing up from behind cover when I access suit functions. Some of the graphics mods overdo it and lack some features only Crytek can add while the remaster is a frustrating PC port of a cutback console port of a great PC game.

How to do this yourself with no mods:
This may not work if 7zip doesn't allow editing the pak/zip file but if you'd prefer not to download anything then here is the method to do it yourself. Make sure to backup the gamedata.dat file mentioned.

• Navigate to CrysisRemastered\Game, this is the folder where you installed the game.
• Rename gamedata.pak to gamedata.zip and open it with 7zip (don not unzip)
• Within 7zip navigate to CrysisRemastered\Game\gamedata.pak\libs\Config\
• Open the file named 'defaultProfile.xml'
• Go to the section 'singleplayer' Change the lines:
  
  <!--action consoleCmd="1" keyboard="f5" name="save" onPress="1" /-->
<!--action consoleCmd="1" keyboard="f9" name="loadLastSave" onPress="1" /-->
<!--action consoleCmd="1" keyboard="f8" name="load" onPress="1" /-->
  To:
  <action consoleCmd="1" keyboard="f5" name="save" onPress="1" />
<action consoleCmd="1" keyboard="f9" name="loadLastSave" onPress="1" />
<action consoleCmd="1" keyboard="f8" name="load" onPress="1" />
  Or optionally change to:
  <action consoleCmd="1" keyboard="f9" name="loadLastSave" onPress="1" />
<action consoleCmd="1" keyboard="f6" name="save_quick" onPress="1" />
<action consoleCmd="1" keyboard="f5" name="save" onPress="1" />
<action consoleCmd="1" keyboard="f8" name="load" onPress="1" />
  This will enable all the forms of saving and loading. You could also change the keybindings but duplicates won't work, only one will function.
• Save the file
• When prompted by 7zip to update the archive click yes.
• Rename gamedata.pak to gamedata.zip
• Load the game in devmode, this can be done in various ways but the easiest is to run the file named 'crysis_DEVMODE.bat' in the CrysisRemastered\bin64 folder
  

Pls help:
I'm sure someone with a little bit more knowledge in this could make something much better from this but I literally did this in 7zip and notepad and just searched for the word 'save' and 'load' across all the files.