The ocean views are a big draw. While it's not the only coastal hot spring town, there aren't a lot of towns with as any hot springs and hotels as Atami has in such close proximity to the coast. That makes it easier to find a place to stay, and Atami is also big enough to have a bit of a downtown area.
Like kamahen said, Atami's location and transportation logistics make it a convenient weekend getaway for Tokyoites. It also makes a good place to stay if you're traveling around the Izu Peninsula. Most of Izu's tourism attractions are morning/afternoon places, and Atami provides more nighttime amenities than the other towns in the area. Starting sightseeing in the morning, heading back to the hotel in Atami in the evening, gazing out at the ocean view from your room or bath as the sun goes down, then eating some locally caught fish is a pretty popular tourism plan.
uIfve been to a few onsen, really enjoy the coziness and the old style buildings. Kinosaki, Takaragawa onsenkaku, Himi onsen. Even ones without the old feel like noboribetsu and what I imagine Kusatsu to be like, has the vibe.v
To most Japanese people, Atami has a vibe too. It isn't the historical "old Japan" atmosphere of the places you mention, but more of a retro Showa-era feeling. That might not be as appealing to non-Japanese travelers who want a ore classical JAPAN! atmosphere, and even among Japanese travelers Atami attracts an older clientele, but it's something some people like.
uWouldnft tokyoites rather go to Gunma for their onsens? If they want Mount Fuji, therefs hakone and the 5 lakes. If they want beach side, therefs shirahama and Chiba?v
Gunma is in the mountains, so if someone wants to go to a hot spring, have a nice view of the ocean, AND eat fresh seafood, Gunma is only going to check one of those boxes, but Atami checks all three. Likewise, if someone wants to go to a hot spring, see mountain scenery, and do some hiking, they're going to pick Gunma over Atami.
You're right that Hakone provides better views of Mt. Fuji than Atami does. I've never heard anyone say they're taking a trip to Atami because they want to see Mt. Fuji, though, so Hakone having the better view of the mountain isn't a mark against Atami (just like Atami's better ocean views aren't a mark against Hakone).
Shirahama has fewer hotels and onsen than Atami, and the ones it does have tend to be smaller, so bookings, especially for large groups, are comparatively easier in Atami. Shirahama/Shimoda is also about an hour and a half farther from Tokyo than Atami, which also means an extra hour and a half to get back to Tokyo on the day your trip ends. Coming from Tokyo, you have to go through Atami to get to Shirahama/Shimoda anyway, and since there isn't much to do after sundown in Shimoda anyway, a lot of people think it's more convenient to stay in Atami and visit Shirahama/Shimoda as a half-day trip from there.
As for Chiba, it doesn't have any towns that have a strong reputation for good hot springs and the level of infrastructure and convenient transportation access that Atami does.
Honestly, though, for Tokyoites, "Atami or Gunma or Hakone?" isn't such a weighty decision. They're all viable overnight trips from Tokyo, and while you can get crazy-expensive if you stay in their finest hotels and tack on the highest-tier meal plans, they also all have plenty of budget-friendly hotels and restaurants too. None of them represents a once-in-a-lifetime travel experience for Tokyoites, and it's pretty easy to hit up all three in the same year even. I've taken hot spring trips to Atami, Hakone, and Gunma (Ikaho, Kusatsu, and Akagi) and liked all of them, but for different reasons.
All that said, from the other hot spring towns you mention as feeling more attractive to you, a lot of them are in the mountains and have classical/historical architecture than Atami does, and if the seaside and Showa vibe don't appeal to you, you might not enjoy Atami as much, especially if you're in the mindset/situation where going to Atami means giving up an opportunity to go somewhere else. I think that's oftentimes the situation for non-Japanese travelers, which is why Atami isn't as popular or satisfying for foreign tourists in Japan, but for Japanese travelers, it's still a popular place.