It is really easy to map a compass to a vibrating belt (someone did it) and learn to feel it instinctively.
But for useful things: Either the sensor is too big (like radar, might be changing right now), or the bandwidth to high (most camera things) to sense through the skin or ears (you don't want to occupy a whole eye), or it's just not useful (e.g. infrared thermometer to vibration for remote temperature sensing).
Our body is made for this world, it's pretty good at sensing it.
An electromagnetic sense could be so much richer than a compass direction. Wonderful to sense the actions of machines and electrical systems around us, and have a boost to navigation. Furthermore ideally I think we'd need to upgrade our internal systems to have a better intuitive understanding of frequency-domain phenomena at high (megahertz) frequencies.
Ionising radiation is scary partly because you just can't sense it. Internal scintillation crystals + additional optical tissue based on our current design = a whole new sense. Obviously with a little redundancy and care of placement you could get alpha, beta, gamma and an energy spectrum. Useful for spacefarers, perhaps.
Interfaces to technological devices would be nice to have. Perhaps you wouldn't call it a sense but in a way it's so much more. Wouldn't want to be stuck with an obsolete protocol in my brain implant chip, though - the format wars would be intense.
The brain interface is more distant in the future and more scary. I prefer going through a normal sense for simple values. You can ignore, rationalize and somewhat isolate a sensation, but not a thought.
I don't know about that latter part. There are so many animals with really amazing senses that make humans seem primitive in comparison.
Wouldn't you love to have the nose of a bloodhound or the ears of a bat? Or be able to see the infrared or ultraviolet spectrum?
That's not to mention all the many senses that are conceivable but not present in any known species, or the potential for synesthesia, like tasting colors or hearing smells, etc. Humankind has a very, very long way to go to fully sensing the world around us.
Many of these untapped senses are super useful, too. An enhanced sense of smell or sight could very well save your life -- some dogs, for instance, have been able to smell disease conditions (not to mention their renowned tracking ability which has saved many lives and solved many crimes).
Synestesia is not so interesting to me, that is more an involuntary mix of senses, where through association I can have it voluntarily and on demand. Do you have more exapmles of "all the many senses that are conceivable but not present in any known species"? Especially some where one could utilize small sensors?
To feel like a real sense this has to be augmented though, not just a heads up display. Only with the (low latency) feedback from movement to a change in the sensation can we learn to use it like a sense really well¹.
¹citation needed, but there is one somewhere
The downside was that some of them experienced nausea and disorientation when they discontinued use of the device.