I think that the warmth of many of those older records, among other reasons, is partly due to the fact that they didn't have as much low-low end. That was for a number of reasons, some of them related to the equipment used to make it and some of it purposeful due to being delivered on vinyl. Because of that, they could get away with more of what we now refer to as 'mud'.
The mud frequencies are very powerful, and if they are there in addition to low lows, it gets nasty. If those low lows aren't there, you can use more of those 'mud' frequencies which without the confusion of all that low low end will start to move more towards warm than muddy. Well, maybe still muddy to modern young ears who have never heard anything but super-scoopy, smiley face mixes I guess. But still, they can add a powerful warmth that sounds lower than it really is.
With limits on the super-crunchy presence frequencies (partly naturally provided by tape I guess), you end up with a well balanced mix that can sound warm, not harsh, well balanced, and which can be well reproduced on most any system, and that should be hearable at low volumes and not get boom box sounding when it's cranked up more.
That's my theory anyway, but it's not one that most folks would care about today where being both crushed and as faux loudness button loud as possible is probably more important to more people.