Roofing is tough because the samples you see at a supplier or from a roofing contractor often times will look different once applied to your build. The samples themselves are from a different lot than what is supplied and can have variations. If you, and im sure you've already been looking, drive around various neighborhoods, you might see a combination you like similar to what you want on your build. Might not be the same brand or shingle type, but ask for sample packs from your supplier and take them along with you and see if you can come close that way. Here in the desert there are a lot of greys in shingled roofs with the theory that it reflects better than dark keeping the attics cooler. My experience roofing is that once the heat is transfered into the shingle themselves, color has little to do with it. Anyhow, i've never seen any grey tone that appear white but lighter colors appear when the shingle is more monotone, ( the same color granules throughout), compared to shingles that have various colored granules around the edging ,(usually darker highlights), around the perimeter of the tabs. The timberline series that you speak of generally have these highlights as they are a dimensional shingle and heightens the effect. You might also check out elk roofing similar to the timberline gaf for other grey options.

I've done shingle roofs, metal roofs, and rubber roofs. The problem free roofs are the metal roofs. Why? People have a strange desire to get onto a rubber roof, and they do not realize the damage they are causing. 100% of the rubber roofs, had an injury from someone being on the roof. (high heals!, sitting on a chair to watch fireworks, etc). Shingle roofs? Shingle heat and fall out, or they get damaged, etc. (homeowners in the north have to shovel snow off shingle roofs). The metal roofs with a nice pitch shed snow fast. My lastest love...is a cold roof, where there might be two roof decks , and the insulation in the attic space can be packed to the gills. Imagine, But, for a mobile home, it's a metal roof elevated slightly above the insulation, with spaced out 2x4's (or such) provide a 1.5 inch air flow. Special framing at the ends(with metal screens), allow air to flow under the deck and water barrier and over the ridgid insulation. Works great in cold areas of the country.. Unnecessary in warm ares of the country.

I would go with charcoal gray or black.

This is a decent article about exterior whites that architects favour that includes this color chart below. https://www.gardenista.com/posts/10-easy-pieces-architects-white-exterior-paint-picks/amp/