SpaceX's rocket technology is completely different from the legacy rocket tech.
The most "legacy" item of the entire SpaceX technology base is the Merlin-1D engine, 9 of which are used in the first stage of the Falcon 9 and 1 of which is used in the upper stage. The Merlin-1D's distant ancestor from the 1990s is TRW's TR-106. When the Space Launch Initiative was canceled, Elon Musk hired the laid off enginers at TRW to continue to develop the TR-106's design. Over the next 12 years it became the quite different, but related, Merlin-1D.
The TR-106 itself was a clean sheet design that owned nothing to previous engines. It was an engine designed around a cost-effective construction and abalative cooling, which made it significant cheaper 0 and also completely alien - to its predecessors and alternatives.
This distinction is important because in the rocket world, new engines are an extreme rarity and generational design inheritance is the norm. For example, the RS-68 engine used on the Delta IV was directly designed as a lower cost / fewer parts relative to the Space Shuttle Main Engine RS-25D. They're different engines, but they're basically father and son. Basically whereas the RS-25 was optimized for reuse (and thus, could be pricier), the RS-68 was built around lower cost and expendability. The RS-25D in term, is a direct deriitivitve of the never flown post-Apollo HG-3 engine, which itself was a direct derivitive of the J-2 engine. The J-2 was used on the Saturn V's S-II and S-IVB stages, and on the Saturn IB.
So when we talk about Beoing's upcoming Space Launch System, which has RS-25 engines, it is a very and factually accurate thing to say "its engines are owed to Poject Apollo and government investment". But considering the technological obsolescence of much of the rocket, that's not in its favor. The approach of SpaceX, which is largely independent of any legacy government-owned designs or government investments, is the superior and more economical way forward.