We have always had decent access to MLC and Sherwin spray and wipe and BAC locally with both having rock solid in-house matching especially with regards to MLC because it happened to be coming from a fully equipped autobody supply so for them to match a wood sample was an utter joke compared to their daily matching faded auto finishes. Sherwin on the other hand in this area has always been hit or miss. They were more apt to pull the closest chip match and ship it on a first try and then you'd have to push them to get it closer.

Auto body supply dropped MLC (too much work for not enough volume) and it has now gone to new distributor 1.5 hours away that can take days to do a stain match.

We are constantly having to push a stain a bit one way or another and Im wondering if anyone has any input on a hand full of core colors of universal tint (and brand) for us to have on-hand to do fairly basic color shifts. Killing reds (+ Green) is our most common. But I'd like to have a reasonable quantity of a core group of colors to make failry basic shifts.

Our work will always be best suited by making these adjustments at the stain level. We tone top coats fairly regularly but the default for the bulk of what we do is to get it at the stain level and then just clear. We dont do a ton of very high level matching/toning/glazing.

If you had any input on a core base of tint colors to keep on the shelf at all times it'd be much appreciated.

Before Aqueous coatings came on the market in force, most wood finishing colors were pretty standard. Your earth colors were VD brown, burnt and raw umber, burnt and raw sienna, yellow ochre,
And a few other garments from differing mining regions around the world.
The most common vehicles used were Japan or linseed oil, and alkyds, with appropriate reducers thinners.
What you seem to be asking for sounds like your wanting tints or shading materials, reds, greens, yellows, blues, plus black and white to lighten or darken,?
Since their is such a wide range of pigment suppliers or manufacturers, it really is up to you to search for what you need, be it for aromatic or aliphatic based stains or aqueous based.
Look at Mowhawk, sheffield, delta, for starters, and for more industrial pigments Huls.
In all honesty, you would be better off making your own stains from scratch.!! But that takes time to learn.!
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I've been using 824 and 844 UTCS and Industrial colorants for 25 years to hand mix and match stains. I even spent 6 years as the color guy for MLC in one of their largest distributors. I always view color in terms of red and green or hot and cold. Raw umber and burnt umber are the base color for majority of stains. Raw umber is greenish brown where burnt umber is reddish brown. It can get complicated from that point. Different yellows to brighten, or soften a color, green to kill red, or combine yellow and black to make green without turning the grain green. Then you run into using dyes to keep clarity and irradesance instead of pigments......it goes on and on until your eyes roll back into your head. My advice......core colors for in shop tweaking would be Raw Umber, Burnt Umber, Lamp Black, Organic Yellow, Yellow Oxide (or Raw Sienna), Touludine Red, Burnt Sienna, Phthalo Green ,Titanium white. You can shift anything anyway with those colorants. For instance.....Van Dyke Brown is literally 4 parts burnt umber to one part lamp black. The only added color, but not essential would be Blue (to shift purple tones). That is so rare though and costs so much its not really worth it.