Amaan Kulshreshtha Hrrrm... well let's go down the list of where it could be failing.
Are you able to go at the output with a document inspector? See / make sure all the rules you think should be applied ARE being applied?
Could be a specificity issue for font-face. Document inspection would reveal that instantly.
Do other attempts to use font-awesome directly work? Are you sure the fonts in question are even being loaded?
Are you using the silly 'empty italic tag' method anywhere on your page? If not you might consider loading FA's font files directly bypassing their rather hefty (and somewhat pointless) stylesheet.
You might also want to verify the path -- as a rule of thumb I HATE using ~ or up-tree links in includes as it usually just means a sloppy directory format. Try moving the font-awesome.min.css to the same directory as the rest of the CSS so you don't have to absolute path it.
It could also be a XSS blocking issue, where browser security is screwing with you -- part of why I dislike using their CDN and prefer to self-host files if I'm going to use them. Relying on someone else's CDN is always a bit of a risk and quite often a pain in the backside to diagnose.
Also are you SURE the weight and style matches the font declaration? One thing I ALWAYS do is use the "condensed' font property so I'm always 100% certain of EVERYTHING being declared. Good rule of thumb, don't trust the default line-height, you change the font-size change the font-height, don't trust the LIE about 'metricless line-height' inheriting 'properly', and by the time you type out font-size, line-height, and font-family you would use less text just doing the one-liner.
Hence:
font:normal 125%/120% "'Font Awesome 5 Solid";
MAY actually fix it. Webfonts can at times up and decide not to apply if you're weight and style don't match those declared in the include.
A final thing to try is to add the wasteful empty italic method to your template and not try to DIY in the CSS. I wouldn't deploy that (I dislike excess DOM elements since I actually USE the DOM) but it could at least help diagnose that YES, font-awesome is being loaded.
I think of anything else to check I'll add it here. There are a LOT of places this could be going wrong for you.