The best way to beat nostalgia is to actually go and re-experience the old thing. Which is actually possible for gamers, unlike many old experiences.
Personally I've been playing FH3, FH4 and FH5 in parallel over the last year, and I can tell you for sure that the clunky driving physics and lack of surface texture differentiation on dirt in FH3 ruin the experience for me compared to especially FH5. Sure, it probably does have a more interesting map, but it doesn't matter when the driving is clearly worse.
I wouldn't even think about trying FH1 today. Back when I played it on Xbox 360 I already struggled with the physics and hated the barriers everywhere ruining the open world experience. Maybe I'm not American enough to enjoy the story, but I just didn't click with it at all...
I do think racing games often get looked at with rose tinted glasses and don't hold up too well, particularly ones that make some effort to be somewhat realistic. Back when we just had Gran Turismo 2 or whatever I think a lot of used some imagination/creativity in our brains to pretend it was more realistic so I feel like we often forget how poorly some of the driving mechanics age. Instead we focus on how cool it was to learn about cars we had never seen before because they weren't available in our country or whatever, or learning cool racetracks from around the world we had never seen before, and so on.
Of course, with the internet and everything being on YouTube to experience and stuff, some of that "magic" is inevitably lost a bit as we grow older and technology advances further. The wonder of discovery is harder to find.
As for FH1, I tried it not too long ago and yeah it didn't feel great, but not miles worse than FM6/FH3 and you could certainly feel the DNA. The main thing I missed about the game was the roads, which I felt were the most engaging in the series so far, even if we couldn't go off road and stuff.
The road up Red Rock Canyon is almost as good as I remembered it, and my favorite in the whole series by a clear margin.
Here's another thing: this is the fifth iteration of a game we're talking about here. When was the last time the fifth version of something ever felt as thrilling and full of promise and potential as the first? And what can you even do to keep that novelty going with a franchise -- any franchise -- where the creators are constantly trying to negotiate future plans while trying to balance the two loudest, most incompatible negative feedback loops of they changed it, now it's bad and they didn't change anything, now I'm bored?
It is certainly a challenge, and one that is kinda unique to racing games. Racing isn't really something that can be totally revolutionized before it becomes too much of a departure, so the key is in the subtle details and how things are leveraged, content and so on... which to be fair is kinda tough to do, but I know I've seen good ideas here and elsewhere for the series that would spice things up without requiring a whole revolution.
I see a lot about the racing genre becoming stale or dying or whatever, but it depends a bit on what you want. As you said, we are so far into things that it's tough for much to feel exciting or "new" or whatever so that's tough to find in any genre, but of course there are areas that are going well. For example, if you want competitive races then it is at a really good point right now, because of high speed internet and multiplayer being easier to use... unless you want competitive single player races, then you really feel the lack of AI development as it feels like we are still racing AI from the early 2000s. Then you look at progression and it's mixed up a lot with some stuff stuck behind grinds and others given out to the point they have no value, Career/story progression in racing games is also weird, with many losing any sense of progression with an over-reliance on freedom of choice, and some forcing you into a "anything other than a win is a loss" mindset while others completely throw results out and just give you money and prizes regardless of how bad you do. Even within Forza Horizon you see a bit of both sides to that.
I feel like we might be coming to a bit of an end for games that try to straddle the middle line as people become more divided on issues, which is something we see in the racing genre a bit. In the Forza part of things, we see it constantly with both series, where the broad playerbase they have brought together are all pulling in opposite directions.
"More multiplayer modes!"
"No, multiplayer is stupid! We need a story for career mode!"
"A story would be predictable and cheesy, we just need more character customization!"
"No the game needs more drifting!"
"No one cares about the story, we just want the new Koenigsegg CC850!"
"No, hypercars are boring we need the new Civic Type-R!"
"You are both wrong, we need a minivan! Quirky cars are the best!"
Unfortunately the racing genre is likely to become even more niche as time goes on, so it's unlikely we have new franchises popping up to focus on individual sub-cultures at the quality level of titles like Forza and GT.