I'm not going to argue that the old bass tones sounded better, that's subjective based on what your tonal goals are. For me I prefer the saturation and power amp overdrive along with speakers being pushed harder that was often a result of inadequate equipment. And can that sound still be achieved now? Short answer yes, because you can get all that gear still, or often times recreations. Look at the guitar side. You have many high quality spot on replicas of old amps (thinking of Marshalls here, Germino, Metro, Rockitt, etc). As far as getting that sound through emualtors and OD pedals and what not...I honestly can't wrap my head around that. We have known quantities that will provide you exactly what you're looking for, but instead people spend all this time and money to put these "tonez" in a stomp box. I get it, people want portability and not want to lug all this gear. But the way I look at it is that's taking the easy way out, and you're not going to achieve these things with that mind set. You're going to have to work for it, packing that gear, maintaining it, etc. No shortcuts will really get you there. "Hey, if I throw this pedal in front of my xyz with my xyz cab, it'll sound just like so and so." Except unless you're using a cab with a totally flat response, and just a power amp with no colouration, it won't. If you're using a Kubicki into some Genz-Benz into a ampeg 4x10, no pedal you put infront of it will make it sound like a Jazz Bass run into a wall of cranked Marshalls and 4x12s. I'd also say I'm mostly speaking in regards to live sound here, because as many have pointed out, there is such a rabbit hole of variables when it comes to studio recordings. Now for recording purposes, sure, if you run your bass DI into the board, and then use the best emulators for the micing, cab, amp, board, whatever, that may work.
And as far as maybe not achieving those tones you're after with a SVT, well, I'm a bit biased, but I'd say that's likely because it's not really the amp that was held as a standard for pre-1970 recordings. If you're after the sound of British bands from that period, you should be looking at Hiwatts and Marshalls, which I think and SVT and 8x10s are quite a far cry from. A lot of people will say that SVTs are the sound of classic rock, but given they weren't out until '69, they aren't the sound of earlier classic rock (nor many of the big bands in the early 70s). Or, if you are using an SVT, not cranking the preamp gain, but making sure that power section is reaching distortion (maybe a bit of a volume issue with an SVT). If you're looking for power amp drive, I recommend lower wattage amps and efficient speakers, which is the route I go.