Good morning,

First and foremost, my thoughts are that I shouldn't.

Now the question:

I recently purchased a set of Apex EC-7 wheels, and they are fitted with BFG g-Force Rival S 1.5. However, they have a date stamp of 2019. I have no intention of using them on the track, but could I street drive them? I'm a truck driver and only get to enjoy my car four days out of the month, so they won't be used much until I decide on a different tire.

Don't be afraid to be mean, I'm a old salty vet that has no feelings left.

Inspect heavily, does the rubber appear like they sat outside in the sun for 7 years.
Any cracks, even minor starting?

If you had bought them new and they were in a climate controlled (more or less) basement
for most of that time, then I'd say sure, for street only.

But keep in mind, overall traction may be significantly reduced, vs the exact same new tire.

Check them for obvious signs of deterioration or damage, if they hold air they are good to go. I wouldn't go set any land speed records with them but daily drives should be fine. I would call them "taxi tyres", rock hard so they last a long time, comes with a good dose of limited grip, driver beware on that side of things, even worse in the wet. Just because they have tread doesn't mean much.
Anyway, you said they won't be used much till you decide on a new tyre.

The guy said they were in a storage unit but in too may ways I don't trust the unknown. I just really will use them for an around town tire for now. At this point I may look for some takeoff with a history. Again no track just out and about.
Thanks for the think tank.

Like others have said, check for signs of age cracks. Run them a few miles on local streets (not highway speeds) and check again for cracks, blisters, or signs of tread/belt delamination. Drive about 15 minutes on the highway and do another round of checks. If everything looks good, run them to the cords.

Skip ditching tires only to buy take-offs, just keep your eyes open for discounts and get a new set. Discounts are available often through TireRack and also places like Costco. You can even get RE71RZs through Costco , though you have to order through the Automotive Dept. and they mount and balance as part of the price. Lastly, if you are a USAA member , there are special deals on Goodyear tires under the " Perks " section. If you go to Tire Rack use the link here on TMO.