Here's my picture.

Let's say you want to use helium as a lifting gas. Hydrogen can be used too, the numbers don't really change. Helium has a molecular mass about 14% the molecular mass of air at the same conditions (pressure and temperature).

Imagine now you build an airship the size of the Hindenburg. About 200000 m3. Each m3 provides a lift of very nearly 1kg, so you can lift with such an airship 200 tons of stuff, but that includes the balloon itself, the cables, everything.

Now imagine the balloon has folds, like an accordion. As it goes up, you expand the balloon until it becomes some sort of very long and very fat sausage. It could even be mile-long.

Where will the balloon stop ascending. Let's say you design the internal pressure to exceed the external pressure by about 1% at sea level. The balloon stops roughly where this 0.01 atm pressure is about 7 times higher than the external pressure (because helium compressed 7 times more than air has the same mass). That happens where the atmospheric pressure is about 0.0015 its sea level value, and this happens at about 50 km height [1]. You could say you are half way to space.

Once you get to this point, you need another source of lift. This is airplane type of lift, except that you don't need wings, because the entire balloon can be tilted slightly upwards and function like a gigantic wing. In order to not suffer from the tyranny of the rocket equation, you need reaction mass. For this you have an outer shell around the balloon that works as a funnel. You scoop whatever air is at that altitude and direct it to inside of a combustion chamber, and you push it at high velocity on the other side. You accelerate slowly over many days. Maybe during the day you use solar panels to heat up the air, and during nights you use hydrogen that you have onboard, and which has a huge energy density. On second thought you might even use hydrogen as a lifting gas, so you have plenty of it at hand.

As you accelerate at some point you reach supersonic and then hypersonic speeds. We are used to thinking these speeds are very destructive to the airframe, but at extremely high altitudes, where the air density is, let's say 1/10000 the density at sea level, maybe they are not so destructive. As you pick up speed, some of the lift comes from the orbital motion. Little by little you keep going up and keep speeding up. I don't see why you can't reach 7 km/s.

[1] https://www.engineeringtoolbox.com/standard-atmosphere-d_604...

Critical bits of Columbia's aluminium airframe melted at about 100km. There's about 250x less air at 100km than at 50km. All in all I'd say this is fanciful.