Yes it is the original wiring... So a good low capacitance interconnect might improve things?
Absolutely! You'll see turntables that were marketed as "Quad Ready" back in the day, and they always had really thick (and usually fairly short) interconnect cables. That is because thin/long cables tend to roll off the high frequencies, and for quad, a turntable needs to reproduce frequencies upwards of 40 kHz.
CDs, or in this case Spotify, will top off at 20 kHz. Both technologies extend beyond the capabilities of human hearing, but it goes to show that vinyl has some pretty tremendous capabilities for reproducing top end.
That being said, there are some limitations that are ever-present for vinyl.
1. Limited dynamic range. Digital sources usually have around a 100dB signal to noise ratio. Because of rumble and other noise, most turntables will only perform in the 70-80dB range. That's before you add actual vinyl, which inherently adds surface noise and imperfections that will reduce that figure. Not to mention, most listeners don't have the finest multi-thousand dollar turntables. This means that record cutters have to further reduce dynamic range in the form of dynamic compression and high frequency limiting, so cheap turntables can still track records. This means that very high-end turntables probably aren't even operating at their fullest capacity most of the time.
2. Noise. Yes there will be more pops, ticks and distortion with vinyl as opposed to the clinical perfection of digital. Old worn records can be worse.
The most important thing you can do is clean them, and your stylus tip, really well. I've had records that I thought were beyond help sound very quiet after a thorough cleaning session.
No matter how perfectly clean you get your records, there will always be that occasional pop or click that is seemingly there for no reason. That's because little particles of dirt or voids can be in the raw in the vinyl prior to pressing, or be pressed into it. Then there are the obvious scratches and warps that can occur from neglect and abuse. Good or bad, it's part of the vinyl experience... That's why many refer to it as a "living, breathing media."
(It also sounds like you might have a setup issue, as others have pointed out. If this is the case, it will most definitely increase distortion and possibly surface noise, and/or lead to an out-of-focus kind of sound.)
3. Inner grooves. Obviously, with streaming digital or digital sources from solid state, there is no limitation of physical media. However, with a round disc, there is physically less real estate near the inner part of a disc than outer. To compensate for this, a CD spins more slowly when reading at the outer edge, and more quickly when reading the inner tracks. This way, the data always streams from the disc at the same rate.
Unfortunately, a record cannot do this. It might sound great at the outer grooves, but as the needle nears the innermost grooves, you'll likely notice that it sounds less dynamic, dull or hazy. It can even sound downright distorted if the record is worn, the setup is poor, or the stylus tip is bad. (IGD=inner groove distortion)
You can hear so much difference from beginning to end of a side because the groove is moving much faster relative to the stylus tip at the outermost grooves, and much more slowly near the innermost grooves. This means there is less room to store and extract audio information near the end of a side, since it's playing a far shorter stretch of groove.
A shibata or fine line stylus will do a lot better job reading tightly packed information. This will help to mitigate audible effects, but it can only do so much. Once again, records are cut to play on less than savory equipment, so dynamic range and frequency response are reduced for the innermost grooves before the cutter head even cuts the groove. This way, most any TT will be able to play them back accurately.
So yes, these are all things that vinyl lovers have to live with. Does every record sound good? No. Does every turntable sound good? Again no. Can the right combination make magic happen? Oh hell yes!