i want the spiral to tighten towards the end. It should have the shape of the blue triangle. How can I make this happen? However, the spiral itself should still look natural. The spiral should in the end look like the one in the last picture (become extremely tight towards the end)....
Seemingly you have got an answer which does the trick. Here's another. Basically it's the same - repeating logarithmically scaled copies of a single turn. That's because to get a conical seamless pattern by chaining scaled copies of a simple shape one must scale the copies logarithmically. That's a math fact.
But I have an easy to make approximation for the turn.
Draw an ellipse
Split the ellipse with the scissors to 2 halves at the waist nodes. Drag the bottom half narrower so that only the right side moves
Select both halves, rejoin them by pressing Ctrl+J. Make a scaled copy. Use Object > Scale > Uniform > Copy and press Ctrl+D to get more copies. The scaling factor in the example is 80%. In my example there's the original + 9 copies.
Build the spring by moving the copies. I didn't try to place the copies automatically. I moved them one by one. To do it perfectly select a copy and change temporarily to the white arrow by holding Ctrl. Drag a node to snap. The rest of the shape follows if you have not clicked anything else. Have Smart guides and Snap to point ON for exact snapping. Join the pieces by Ctrl+J.
Make a longer spring by inserting a scaled version of what's got in phase 4. The scaling can be exact. In my case I scaled at first to 80% (=the same as for scaling the turns). Then I clicked Ctrl+D nine times to get one step smaller copy than the last turn of the tiled spring.
To get a little clearer narrow end: Join the parts, select a narrow stroke and make the starting end wider with the Width tool.
In theory the narrow end cannot be made perfectly sharp with this tiling principle because it would need infinitely turns. One can in practice stop when the tip is visually sharp enough.
Colorizing with a gradient may create more 3D-like illusion. But it's useless for thin strokes like this