Well if you want to plant right away then yes you are limited to what is in stock at this time. And yes you can spend a lot of time identifying kinds you like only to discover they are effectively not on the market. Filtering for big sizes reduces your options additionally to a significant degree. Especially these days, when there's a focus among consumers on flowering annuals, vegetable plants and other edible gardening goods. And not on woody plants - some outlets have had hardly any woody plants on display in recent years, let alone big specimens.

Despite this overall trend multiple different companies specializing in big trees specifically have been in operation in our region. If you have not done so already you should be able to find their listings in the same places as those for independent garden centers and other sources.

"Mature(ish)" - what does that mean?

Right now is when the better local independent nurseries receive their current season stock of large woody plants, so now is when you are likely to find the best selection of trees. I spent last weekend at my local nursery helping two sets of design clients pick out maturish trees for their gardens - multi-trunked stewartias, a 10' tall paperbark maple, a large 'Bloodgood' Japanese maple, 'Thunderhead' black pine, 'Frisia' black locust, a big katsura, Japanese snowbell, etc. All great trees for a PNW garden and none of these are what I would consider "starter" trees - they all have a number of years on them :-)

It comes down to what specific trees and what specific sizes you are looking for, but often just strolling through the tree section of the best local will produce satisfactory material from which to choose.

Isn't there a nursery north of Everett that specializes in larger trees? I don't know the name. Olympic nursery in Woodinville also used to do some fairly mature(ish) trees.

Big Trees is located in Snohomish. But I believe the OP lives in the Portland area so a bit of a trek to drive so far to find a few larger trees. I'm sure there must be something similar in northwestern OR.

AAMF, a quick Google turned up Big Trees Today in Hillsboro.