You are correct that reinforced concrete will corrode, generally the service life of reinforced concrete is around 50 years if no special measures are taken. This is because the initial alkalinity of the concrete, which protects the steel, will be neutralized over time. This is the same for modern and for Roman concrete.
But many methods are available both to extend life, and to monitor the health of reinforced concrete. The failures that do occur are typically due to negligent maintenance.
The Romans didn't use steel reinforcement, of course, they just used massive concrete structures. We can do the same today, but it would mean no skyscrapers, no long span bridges, etc.
To be able to build the Pantheon, at 43 meters height, they used walls with a thickness going from 6.4 meters at the base down to just 1.2 meters at the top. In addition they reduced the density of the concrete by almost 50% towards the top, by using very light aggregates.
In comparison the Burj Khalifa at 828 meters heigt has less than a tenth of the wall thickness at the base, just 0.6 meters of reinforced concrete wall thickness in the hexagonal frame.