Well, the risk now is that if the present wheel is so far out of whack that it's managed to bust 3 spokes it's likely so poorly tuned that a lot of the other spokes are well on their way to breaking. So even tuning up the present wheel would likely not result in a trouble free future.

I hate to say it but as I said already this shop didn't do right by you starting with the FIRST spoke by not noticing that the wheel was poorly tensioned. I don't see any way around it, you SHOULD get another wheel. But I think I'd get it from another shop. And bonus points to any shop that suggests pre-tuning the wheel instead of just stuffing it on and sending you on your way.

As mentioned above a 32 spoke decently "top up tuned" machine wheel should only add about $25 to $30 the purchase price and such a wheel should/would be as trouble free as a hand built that used the same components.

I've done about 1500 km's on my last set of hand top up tuned machine wheels. The last time I checked the tensions were all still spot on. And the wheels have not required any truing at all during this time. Granted 1500kms is just getting nicely "broken in" for a decent wheel but it shows you that you can expect a long time between any future tune ups if it's done well from the beginning.