No need to worry about leaks. Definitely have the contractor make it right though as it will affect your warranty. Also, make sure your contractor registers your warranty. It seems that warranties are being sold and then never registered. It was probably a communication issue as opposed to an attempted scam. Seeing as the contractor is willing to make it right I say you are in good shape.
I don’t know about where your at but here in NH more than half the roofs we replace have no underlayment and most have the three tabs as the cap. The older three tab roofs are one thing but the roofs with arch shingles slapped up in developments should have a hip and ridge shingle or it will fail before the roof itself as pointed out in the pictures with the granules missing and cracks all through the cap.
We cut open almost all our roofs and install ridge vent, soffits, and block or side gables if present. You keep on installing the three tabs as cap and your felt paper and they will fail the same. but I guess it’s not your problem after the first 10 years huh? which is how old two of these roofs are.
Everyone is welcome to an opinion I guess. No question, specific hip and ridge shingles are the best. However, there are millions of roofs out there with tabs from 3 tab shingles utilized as cap and they are working just fine. It is very, very rare for us to find a roof failure that is from the cap. And when that happens, it is almost always an install issue, not a material issue.