Over the years I've had to get acquainted with how something sounds in phones as opposed to speakers. I used to be quite shocked when I'd hear something that seemed au fait in phones on the speakers. Now, I don't. There's just a way one learns to listen. There's a way I can get the bass to duck under drums and get the kick to plop on the bass so I can hear both or shake the guitar or whatever and at that point I know it'll be OK on speakers.In some ways, this is the best position to be in.........but that's dependent on you and your tastes. In a sense, what you're saying is that you don't really know what sound you want from the bass but you'll know it when you find/hear it and what you currently have isn't it.
My acoustic bass guitar has an input and because it's fretless, I sometimes use it as an approximation of a fretless bass. But what a struggle I have to get a sound that accords with what I like ! Which brings me to my main point ~ experiment. Your experimentation may involve you being fussy or not fussy. I think I searched for a long time [a couple of decades, actually} for a bass sound that I was happy with. Part of that was the bass I had at the time, part of that was the amp, part of that was the way I play and part of that was that I didn't really know exactly what I was looking for. When I played live I never really worried about my bass 'sound' because I was always in the hands of a sound engineer who would do this and that from the desk or some instrumentalist or singer that would have very definite ideas about how the bass should sound. I like a more twangy bass but more often than not, many of the people I played with wanted ultra deep bass with no treble whatsoever ~ especially once I'd bought a 5 string. Prior to that I just plugged it into whatever amp I used and played ! If sound came out and was deep, that was fine.
So when it came to recording it took me a while to get to that place where I could work out a variety of different bass sounds and utilize them at different times. The way I've been for the last 10 or so years is that I never get to flabby or woofy on my electric bass. I have a number of different ways of recording it. Sometimes I'll just DI straight into my recorder. Or I'll mic the amp {either the bass or a guitar amp}. Or I'll play through the amp but DI the amp. Or I'll simultaneously play through a bass amp and a guitar amp, both at very different settings, then I'll DI from the amps and blend the two tracks. Or I might mic both amps and blend. Or mic one and DI one and blend. Or do a three tier thing where I DI straight in, mic the amp and take the line out of the amp {for some growl} and then blend the three sounds into one. Or I might use my "Sans amp" { a cheap but very adaptable Behringer. I think it's the BD21 or something like that}, either on its own or in combo with any one of the above mentioned. Bear in mind that if utilizing two or more sources, the settings have to be pretty different but having said that, maybe not ! I'll also use a variety of guitar pedals I have or even put it through the effects section of my mixer. It's a similar story of that aforementioned acoustic bass guitar although there, there is more chance of the sound becoming flabby and woofy. But by experimenting, I've narrowed down a sound or two that works well. Compression on it can be good and tight or a disaster.
The point is that there really are any number of ways of recording the bass. I appreciate that to many people "it's just the bass" and as ez willis once so eloquently put it, the sound of the bass is not going to be the deal breaker in whether or not a person likes a song. But I take a different approach when it comes to musical instruments like electric guitar and bass guitar and drums in particular. I think that they are packed with such a variety of tones and those tones, the actual sonics, can actually be a contributing factor in how appealing a song sounds. My oldest son likes a lot of indie stuff and when he plays stuff for me, I always zero in on the bass. And depending on who the band or artist is and when the song was recorded, I hear a juicy variety of bass tones.
But always keep in mind that the instrument, as great as it may sound on its own, is the servant of the song. Bass guitars make empathetic servants but hostile, overbearing masters !