We Know Why Ancient Roman Concrete Lasts Thousands of Years

As for rebar making concrete structures less durable; yes, that's certainly true for steel rebar. The reason being that it will rust, very slowly at first, but once it starts the expansion of the rusted part causes cracks in the concrete which allow more humidity and oxygen to reach the steel, thus rusting faster. This is often called "concrete cancer", and limits the useful lifetime of most modern reinforced concrete structures to between 50 and 250 years (depending on the environment they are in, the forces they are exposed to, and the quality of the concrete they were constructed with).

Concrete cancer can be reduced or even eliminated by using rebar material that rusts more slowly (stainless steel) or not at all (carbon fiber), but these are much more expensive of course. There is room for research on other reinforcing materials, but basically nothing with good tensile strength is going to be cheaper than steel and considering the quantities of rebar we use, cost is definitely a major issue.

The self-healing nature of Roman concrete might also help here, but the chemistry of concrete and rust formation on embedded steel is complex, and without extensive experimentation right now we don't know if steel embedded in Roman concrete rusts faster or more slowly than in modern concrete (before considering cracks).

You can't build sky-scrapers without rebar (i.e. with un-reinforced concrete), but you can build some pretty large structures if you use curves, arches, widening bases, buttresses, etc. The Pantheon is pretty big, built from un-reinforced concrete, and nearly 2000 years old.

You have to adapt your building style to the material you're working with and tall, thin structures depend on the tensile strength of steel; concrete doesn't have much tensile strength, but does have tremendous compressive strength, so your structure will have to be wider at the bottom, although not necessarily wider than it's tall. It's all about directing the vectors of forces in a way that they stay inside the material of the structure, so no flying slabs, upper floors have to have arches or domes supporting them from below (or lots of pillars that widen into a small arch at the ends).

Here is an idea for a technique may be useful for building with un-reinforced concrete: instead of pouring whole walls into a mold, pour "lego"-style interlocking (large) blocks, layer by layer. Between layers you paint the surface with a thin layer of weak but flexible mortar or glue before pouring the next layer. This way you keep enough room for the structure to shift and settle without cracking and you can use the angle of contact between blocks to deflect the vectors of force back into the material. The article mentions that the Roman-style concrete hardens much faster, so that'll work well with this idea (you don't have to wait too long between pours).

I think the problem is the kinds of structures we want to build. The Romans built everything supported by arches, where the loads are all distributed in compression. But to make a glass-walled multistory apartment building with overhanging balconies you definitely need rebar to handle those tensile loads.