Sega had a chance to hold on to enough market dominance to remain as the third console player even after this, but then their fate was sealed at the very instant they decided to put a CD-ROM drive in the Dreamcast instead of a DVD drive.
Dreamcast was released at a bad time, DVD components were still expensive so if they’d included a DVD drive it would have provided some future-proofing, but the console would have been even more expensive than it already was
There is no way they could have put a DVD drive and the necessary playback hardware in the Dreamcast and still sold it for a price people would pay in 1998. Standalone DVD players still cost $600-$1,000 back then. The argument should be that Sega launched the Dreamcast too early, but they were in dire straits and needed to replace the Saturn sooner than later. I’m not convinced they had much choice.
I think the PS2’s success is a lot more complex than “it was a DVD player and a game console in one”. The PS2 also benefitted from the massive amount of momentum built on the PS1, backwards compatibility, a better controller, and much faster hardware.
The lack of a DVD drive isn’t what killed the Dreamcast. I’d argue that the nail in the Dreamcast’s coffin was when software piracy on the platform became trivial.
Please read the linked article at the end, from someone who lived at that time it brings great memories of the console wars: https://arstechnica.com/gadgets/2013/01/how-i-launched-3-consoles-and-found-true-love-at-babbages-store-no-9/
