Where is my End Portal?

Try using the tactic called X-Ray.
See this link for more info.

Just in case the video is deleted, In the video the user digs 3 blocks down, then jumps and places a wooden fences. After this, the user jumps once again, but this time he places a piece of glowstone on top of the fence. Tada, you can now see all the caves and openings around you.

It's possible that ravine generation destroyed the portal or didn't allow it to generate. Without cheating, you don't have many options other than to keep combing through it as much as you can.

"Is there a way I can still salvage this game without having to resort to cheating?"

Yes, there are two options here:

1. Keep looking, but ignore the Eye of Ender tracking. Using @Jonoco98's method may help you, and you can use TNT blocks and pistons instead of wooden fences (Caution: This takes advantage of in game quirk; some people still consider this cheating). The pistons push the TNT blocks into you (if they push into a wall) and allow you to 'see' through the world. Be careful not to blow yourself up. Since you're in hardcore, it may be better to do the glowstone/fence method. Do this a couple times around the area - in a place with lots of blocks - as air blocks will obstruct your vision.

2. There should be 2 more strongholds in your world that you could locate instead. According to this wiki, each stronghold will be 120 degrees away from each other relative to 0,0 - between 640 to 1152 blocks from the original spawn point.

The easiest way to get to a new stronghold is to move away from the current stronghold throwing Eyes of Ender until they change direction. If you want to conserve the eyes, and you trust in my math skills, start looking at coordinates x:-531, y:358. This should be 120 degrees relative to 0,0 according to your current position and 640 blocks diagonally from 0,0. This definitely isn't exact - but should get you in the right ballpark.

You should check out this video on how to find the ender portal. It isn't always near where the eye of ender goes, but it is always in the dungeon. And the eye of ender will always take you to the dungeon.

The basic idea here is to take eye of ender (easiest in creative mode) and throw them while flying. Go in the direction the eye flies toward. When the eye stops moving forward you need to dig down into the ground below until you find the dungeon, which will be directly below the place where the eye stops.

Once in the dungeon the portal will be in the radius around where you entered it.