I think middle stage Pokemon within three-stage lines often occupy a weird space, because you're right, middle stage starters (and middle stage mons as a whole) are generally a lot less popular than the first and final forms of a given three-stage line. They're sandwiched between what is often a cute first stage and a badass/standout final stage of the evolutionary line. Starters are no exception to this: the first form starters are the ambassadors of a given region and are designed to be cute, likable, and to appeal to kids as something that is endearing to them. And then the final form is often badass or otherwise very strong looking in some way or another, because that's the final form and is meant to be a strong Pokemon. Starter final forms often look big, powerful, and standout so as to make them feel like powerhouses that kick butt. Even in later generations where you have arguably more feminine leaning Pokemon like Delphox, Meowscarada, and Primarina, they still have that vibe that they can absolutely deliver in combat and have standout, striking designs: Delphox is a wicked mage, Primarina is a beautiful songstress, and whatnot.
And oftentimes, because of this, middle forms are often not much to write home about. There's not much interesting to look at there. They're neither particularly cute nor very outstandingly cool. There are a few middle stages who are popular, like Braixen, Grovyle, and Raboot, but those are largely exceptions, not the rule.
At the same time, the middle stage is often the key to the success of three stage families and why they can work as a singular evolution family. It is because of the middle stage that a designer can take two extremely different designs, oftentimes at the basic level a very cute first stage and a very badass final stage, and bridge them together. Three-stage lines have that freedom from both a gameplay and a design standpoint where the first and third stages of the line can stand further apart, because they have that middle stage to be the bridge between them. Two-stage evolutionary families do not have this freedom, because the first stage is the first stage and the second stage is the final stage, so design wise they need to be closer to each other and more directly cohesive with each other to work as an evolutionary line. With a three-stage line, you can have an unevolved form and the fully evolved form be more drastically different from each other, and yet they still work because they have that in-between middle form to bridge them together into a singular evolution line.
Like we've been talking about the Sobble line, for instance, and the line's progression works because of Drizzile. Sobble and Inteleon are on pretty opposite extremes from each other despite sharing the same chameleon who can turn invisible trait, but they can be a singular evolution line because of Drizzile. Drizzile serves as that bridge that can connect the two into a singular evolution line. Drizzile's traits are more in-between of Sobble and Inteleon, allowing the designers to bridge the two designs together. Starters in general can have a greater difference between the first and final forms thanks to the middle stage. They can be more extreme in a vacuum and between each other thanks to that middle form to bridge the gap.
In particular starters benefit most from this capacity of design from being a three-stage line because of their specific purpose in the grand scheme of things. They are presented as a Trainer's very first Pokemon, their main friend and partner, who grows with them over the course of the journey and sticks with them from beginning to end. The first form is a cute, likable first stage that is very easy to bond with, and then as they evolve they end up as the final form which is a strong battler and looks cool, meaning they're something a Trainer who at this point of their journey can absolutely kick butt with, as their loyal partner who has been with them through thick and thin, their absolute most trusted partner. From a gameplay standpoint, this also benefits their purpose, as they start off a capable, but not very strong Pokemon early on, but they evolve once and then gain a power jump good enough to allow them to fight during the early-mid game, and then they evolve again in the 30s and are now a fully evolved, powerful Pokemon that can be strong enough to take on anything.
From a design standpoint, starters are designed to be standout for this reason, and so they can have a very cute, lovable first stage for marketing purposes and then a powerful badass final stage that looks cool, with that middle form bridging that gap. So you're right, independent designs with an in-between form to bridge them is a common method, and the way three-stage evolutionary families are structured means that the middle form is that most important component: because it allows designers to bridge two more extreme designs together.
You can't go directly from say, Charmander to Charizard, or say, Frigibax to Baxcalibur, or Axew to Haxorus for instance. The design gap between them is too wide, and gameplay wise the base stat difference is too drastic. But they have those in-between stages that, while not very interesting designs in a vacuum, allow the designers to make those base and final stages that stand further apart from each other yet still form a singular evolution family. This kind of design freedom isn't something they can do with a two-stage family, because the progression is one-time and direct from unevolved to fully evolved. Three-stage families on the other hand have more liberties from a design standpoint in that regard.
