DevL - I agree with your first sentence except when it comes to telescopic sights. As regards your second sentence, "It depends" I hope the below explaination helps.
Eye relief and FOV are first tied to the diameter of the optic you are looking at. For a fixed diameter flat piece of glass, DevL is exactly correct, Get closer, and you'll get a wider FOV, move back and you get a smaller FOV. Traditional Red Dot sights (Aimpoint, EO Tech, Dr. Optic, etceras, all behave this way, and, since their "windows" have zero optical power, you can move your eye closer or further away and still get a clear view through the sight.
When we are looking through an eye piece on a telescope, however, there is the additional constraint in that the eye piece only focuses the telescope image at a certain distance - the eye relief distance of that eye piece. Get too close or too far from this and you'll exit the "eye box", i.e. the volume where your eye should be to get a clear focused image.
Lets put our eye at some desired distance and design an eye piece to focus at that distance. Once done, our FOV is now limited by the diameter of the eye piece. If the eye piece has a small diameter, then you will have a small FOV no matter what the rest of the telescope is does. If the eye piece has a large diameter, then the FOV can be large (if the rest of the telescope doesn't interfere).
The angle between the diameter of the eye piece and the focus distance of the eye piece defines the FOV just like we saw with the flat circular window. However, with a telescope, there are multiple windows to go through and each one has to not interfere with the FOV of the eye piece. Note that once the eye piece is built, you can't get closer and make the FOV bigger. You have to stay in the "eye box" which is centered on the "eye relief" distance in order to see clearly through the telescope.
So, if we briefly compare the ACOG and the ELCAN, you should notice that the eye piece diameters are quite different. Since the ELCAN eye piece has a much larger diameter, it can support a much larger "eye relief" while maintaining an wide FOV. (There are many finer points to eye piece design and human perception that are beyond the scope of this forum).
The ELCAN eye piece has a 34 mm diameter and a 71 mm eye relief. This equates to an eye piece FOV = 2 x ATAN ( 34/(2 x 71)) = 26.9 degrees. Now when the SpecterDR is set to its 1X magnification, the rest of the optics don't limit the eye piece FOV so that the entire telescope can transmit this 26 degree nominal FOV from the scene to the eye piece and subsequently to your eye.
Now, when we switch the SpecterDR to 4X, the erector prism and the front lens limit the FOV to 6.5 degrees nominal. If we make them much much larger, then just like making a window larger, the FOV of these optics could also be increased up to the 26 degree FOV of the eye piece. The resulting telescope however would be huge and very heavy.
Now to address the second sentence....my answer is "It depends".
If we take the 1X SpecterDR optics and redesign the eye piece to have a 4 inch eye relief all we have to do is also increase the diameter of the new eye piece so that the new eye piece still has 26.9 degrees FOV, The new sight will then have longer eye relief and still have 26 degree (nominal) FOV. HOWEVER, if we increase the eye relief and don't increase the eye piece diameter, then the FOV of the eye piece will be lower and so the new sight would have a lower overall FOV.
Now lets consider the 4X SpecterDR optics and again redesign the eye piece to have a 4" eye relief AND lets not increase the eye piece diameter. We will end up with an eye piece with a FOV = 2 x ATAN (34/(2 x 102)) = 18.9 degrees. We might think that 18.9 degrees is still bigger than the 6.5 degrees of the 4X optics so that the system FOV should still be 6.5 degrees but this would be wrong. When you change the "eye relief" of the eye piece without changing the diameter, you also begin to shrink the diameter of the imaging area that the eye piece will relay from the image plane to your eye. Since the front optics have already established an image of a certain diameter, shrinking the amount of this image that gets relayed to your eye shrinks the FOV of the system. The overall result at 4X would be a sight with a smaller overall FOV. If we redesign the eye piece and let the diameter increase accordingly, then the resultant 4X system would retain its 6.5 degree FOV.
There are a significant number of coupled considerations that go into balancing between optics diameters, eye relief, FOV, magnification and optical length, errector prism type, errector prism size, aberation correction, human perception, mechanical ergonimics and suitability for intended use. Mastery of these issues and insight into what choices improve survival and lethality is what careers and companies are made of.