I'm dealing with a 1970-built home and the oven has 6-3 without ground running to the oven. 2 hots and the neutral. Got a new oven coming in and want to make sure it's wired right. I could run new wire, it's not impossible but daunting for a DIY'er. I'm curious about running a separate 6AWG ground wire along side the 6-3 from the panel. I see conflicting info online about how NEC 2014 allows this but then there's NEC 303.3(B) that says all wires need to be together or is that only for running in conduit? The oven wire currently is running up a wall and along a 2nd story floor joist to a junction box, only after the junction box is it in metal conduit with a ground wire bonded to neutral. I plan to remove the junction box and run the 6-3 through the floor to a new junction box up stairs with a 4-prong receptacle. Here's some pictures: https://imgur.com/a/Asq4c9R
You can retrofit ground. See many answers about that - e.g., What does this answer about old residential home grounding mean?
Options include running a new separate ground wire or connecting with another ground wire. However, while you may be able to use a smaller than 6 AWG wire for the ground (10 AWG according to Harper's comment), you can't just connect to a 12 AWG or 14 AWG ground from a 20A or 15A circuit - that's not quite big enough.
As far as all the wires being together - ground is the exception. That's because ground only carries current in an emergency (and then hopefully only long enough for the overcurrent situation to trip the breaker). However, one key to that is make absolutely certain that any neutral-ground bond in the oven is removed. Sometimes a neutral-ground bond is included in the oven but that is only used when connecting without a separate ground to the panel (which should not be the case on any new installation).
As far as why this is all even an issue: When grounding was added as a requirement for most residential circuits, appliance manufacturers (specifically electric ovens/cooktops/ranges and clothes dryers) were concerned that requiring a potentially significant amount of electrical work (e.g., if the wires were not in conduit and the path not easily accessible to run a ground wire or a new cable) would affect sales of new appliances - you would replace it if it was broken and you had no way to fix it, but you wouldn't "upgrade" and you would fix rather than replace when possible. In other words, money. So they got an exception to allow ground to share the neutral connection. In certain situations this provides the same safety benefits of a separate ground wire. But in many other cases it does not. At this point (and for many years now), new installations and changes to existing installation require full separate grounding. But the manufacturers are still resistant to losing any sales due to lack of a ground wire so they continue (legally) to sell appliances that can be connected either way, and in many areas it is totally legal to keep using a ground-via-neutral provided it is a simple appliance replacement, but not recommended.
As far as neutral-ground bond in the main panel, the basic idea is that neutral & ground should be connected in exactly one place. In fact, they need to be connected for certain aspects of the entire electrical system to work properly. But having them connected in multiple places ends up defeating the purpose (partially) of ground wires.
In your situation id just use a standard 3 wire cord on your new range. Remove it from your existing of it is still in good shape and use it. Last new appliance i installed came with the same style ground-neutral strap every other one I've installed had. They all needed you to remove the strap if using a 4 wire cord. ( ironically a drier i just repaired that was wired for a 4 wire cord, while inside the unit i noticed the neutral wire actually split into 2 wires, a white neutral that went to nuetral terminal and a green for the ground that went to the driers frame where the cords ground wire was to be connected. So removing the nuetral-ground bonding strap did nothing). Just check the manual to double check that it comes with the bonding strap pre-installed. Chances are if it's not in a bag with the manual and such its pre-installed. I've never seen one that it wasn't pre-installed. But with electricity i never leave anything to probability so id just double check the manual.