Honestly, while this was an awesome experience in arcades, it was definitely less fun at home after having dropped $50 for it at EB (I picked it up on discount for $20 and still feel ripped off slightly).
This was the beginning of the end for Sega for most. They spent TONS to get that license then put it on this game. Fans of the movies were not happy about only the name being connected and beat em up fans were upset that this game wasn't longer. Nothing new beyond the mini game was added to the Saturn port.
Sega didn't do anything to make the game longer, better, or well anything other than port the arcade game to the Saturn with minimal effort beyond that.
That is where I think Sega started losing fans. Over on PlayStation, Sega's biggest arcade challenger, Namco, were adding TONS of things to their home ports. Namco was raising the bar for home ports of arcade games and they were doing it on PlayStation. Even Capcom added did extra, even if ever so slight, with their home ports of Street Fighter II and such during the 32-Bit era.
Sega on the other hand, well, we got the bare bones port, for full price, of an arcade game that could be beaten with 1/10th investment of the home version. It wasn't just Die Hard. Sega did this with Daytona Racing, Sega Rally Championship (okay, they did a bit more than normal with this port), and Sky Target to name a few.
Sadly, Sega continued this with the Dreamcast all the while Namco and many others were going above and beyond with their arcade to home ports. Even those licensed from arcade game makers and released by another company in the home did more than Sega were doing.
I love Sega arcade games, they are focused fun events but straight ports to the home don't make sense, not at full price.
SEGA arcade games are technically easier to revisit and replay now as the usual method of accessing them is...the easiest method overall if you know what to do and where to get what you need [because most of their catalog hasn't been ported to modern consoles if not at all - again, if you want Fighting Vipers 2 or Daytona USA 2 on consoles, you have to buy Like a Dragon Gaiden and/or Infinte Wealth], at least both in not having to shell out full price for those and infinite credits if playing arcade games through emulation.
That said, it is interesting to hear this account from you regarding how things about SEGA were seen back in the day [as someone born in 2002...so definitely way after SEGA's hardware prime] - caught my eye that you happened to name Sky Target [even if Daytona USA's name was off] as that was one game I kinda know from emulating the arcade version but know that was ported to the Saturn.
There's certainly a few games that had additional bonus and modes, such as Virtual-On having the Jaguarandi hidden boss unlockable, SEGA Touring Car Championship having a separate Saturn mode with a few extra modes [although some Saturn ports in general had a Saturn mode that was allowing you to use a few different cars or an altered story mode], SEGA Rally Championship didn't have an extra campaign mode like SR2 but did have the unlockable Stratos and Time Attack modes, and Daytona USA Championship Circuit Edition had a few extra tracks and cars to choose from while being totally remade from scratch.
It is kind of tragic to think that, while nowadays it is easier to appreciate these games for what they are, it makes sense to think that arcade ports without much meat in them on a console already struggling overseas was really rough compared to PS1 getting games ranging from slightly or noticeably feature rich arcade ports...to games like Final Fantasy VII and Metal Gear Solid that need no introduction in regards to how big they were in single player. And that's not talking about super solid single player games in Saturn that already were sharing house with the PS1.
The Dreamcast did have some bigger and more ambitious games, but it is hard to argue that even if it had even stronger games than what it launched with [and Saturn hadn't stopped the steamrolling they had with the Sega Genesis], the PS2 was just too much to compete with overall. Even the Gamecube, which is beloved in my heart to this day, ended up THIRD behind the Playstation 2 and Xbox in sales - so the Dreamcast even has less of a chance.
Perhaps this is why they have decided to often go with the Yakuza ports route - even to the annoyance of arcade enthusiast wanting to only get the rare arcade game ports, it obviously happens to be easier to sell those inside whole ass games rather than as collections. And not saying that it isn't possible [Marvel vs Capcom Collection is thriving]...but for SEGA arcade titles specifically is far more niche of an idea to bank on outside of Japan.
There's certainly some interesting food for thought in this topic specifically, but it was neat to see someone articulate that much their point of view about that in here - apologies for the even longer reply, you can tell this was quite the fun thought for me as someone that didn't grow in the 90s BUT spent many years messing around with SEGA PC ports from the internet...most of which were based off the Saturn ports 😅
I was born in 1976, I was enjoying Sega arcade ports throughout gaming and it was an amazing feeling to be able to play almost arcade perfect arcade games on the Saturn (I was BIG into arcades at that time too). It just sucked that Sega was wanting $50 for these ports, I understand development is not cheap and people need to be paid but fans voted with their dollars and in cases like this, at least in my circle of friends, PlayStation won almost every time.
I would bring up Daytona (my apologies on getting the name wrong, I am going off memory here) and my friends would point out Ridge Racer. Virtua Fighter 1 and 2 were countered by Tekken. Sky Target was countered with Warhawk (not an arcade port but still even a blind man could see how amazing that was back then).
Adding extras, advertising those extras, would have done so much for Sega but they were horrible at this. They peaked with their advertising with the Sega Genesis and the "Welcome to the Next Level" campaign as it was all downhill after that.
Another area they completely and utterly failed their fanbase was no 2D or 2.5D Sonic on Saturn. Instead we got Clockwork Knight 1 and 2 (mags of the time made sure to point out this was ONE game in Japan) and Bug 1 and 2 (Too). Astal gets a pass here as it was simply beautiful back then.
Sega came into the 32-Bit race ahead of schedule and fans like me thought they did this because they had a new Streets of Rage, Comix Zone, X-Men, Sonic, Phantasy Star, etc ready to go and would drop them closer to the launch of PlayStation six months or so later. Sadly, history shows us, this was not the case and Sega was traversing rough waters with no manning the helm of the ship.
Virtua Fighter Remix was what the original release should have been on Saturn with "arcade visuals" being an option, even make the Remix an unlockable if need be.
I would love to see Sega do a Super Scaler (Space Harrier, Thunder Blade, Super Hang-On, etc) collection even if they only improved the frame rate. The value of a collection would be in the number of games for the price rather than added content or unlockables. Sadly, back in the Saturn days, Sega passed on doing just this with their Sega Ages series and instead we had Working Designs step in to release one with three games on it (After Burner was one, I remember definitely being included).
The Dreamcast had a good amount of 3rd party support, Capcom were kicking up a lot of dust with their support for Sega's last console. Sadly, it was either ahead of its time or was the victim of how Sega ended the Saturn (which did cause EA to snub the DC).
The Dreamcast was a nice farewell to hardware manufacturing for Sega. It is sad but looking back on it, it appears that is exactly what it was meant to be. Sega poured development money into their releases on this console and pushed hard (if only they pushed this hard on the Saturn) but as you mentioned, the PS2 was just too strong an opponent to take on. Sony did so much right by gamers with the 32-Bit generation that it would be near impossible to crack that support Sony had before they even released the next PlayStation console.