Galvanized vs Cold rolled
cannot really be a question, but evidently you meant 'Galvanized vs Painted'

Assuming the handrail experiences only occasional contact with human hands, then galvanized is probably the better choice. Unpainted is not an option outside of arid desert regions. The lifetime will depend on the process and quality of the galvanizing (and therein lies the rub). And of course the local climate.

Ferritic stainless steel is becoming a cost-effective alternative for architectural applications, though that doesn't appear to be your application.

"Everyone is entitled to their own opinions, but they are not entitled to their own facts."

The issue is ongoing maintenance. If they don't keep up with repainting then the bare steel will begin to rust and shed more paint. The Galv would prevent that, at least for a while. And what surface prep did the do before painting? on plain steel it should have been a fairly aggressive blasting. Was a Zn rich primer used under the top coat?
This is why SS hand rail is becoming much more common. I have seen both ferritics and lean duplex being used.

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P.E. Metallurgy

For external handrails and gratings, plastics seem to be a better solution.

Galvanized handrails near a sea waters, collapse in 1 year. I hope it is not too late.

Recently for a UN funded environmental group , I had recommended SS gratings and railings for about 2.5 km length.

"Even,if you are a minority of one, truth is the truth."

Mahatma Gandhi.

As Ed says, it's a maintenance issue. Definitely a question for the owner as it may be an undesirable burden on his/her maintenance staff (if they even have one).

By specifying galvanized/painted, did you mean a duplex coating? That would mean prepping the surface, galvanizing, prepping the surface again, and coating it with a paint or powder coating that is compatible with the underlying zinc. This can be pricey, but it's what I go for on exterior steel fabrications that the architect wants to look pretty. If it was an either/or then you should expect this result every time. Painting is a lot cheaper than galvanizing (especially if the contractor cuts corners and just makes it look nice until the check is in the bank) so for a contractor on a competitive bid it's a no-brainer. So if you do fight it, your client may end up with a change order to pay for the materials they already purchased, the new ones, and the galvanizing.

If the rails are shop-fabricated in sections to be installed on site then fabricating from non-galvanized stock and galvanizing the completed sections, then painting, then installing is probably the best of the possible options.

If the rails are all site-fabricated (I'm assuming welding) then it's a mess anyway. It's difficult to weld galvanized metal, and the fumes produced when you try are toxic. You can try to strip the zinc locally before welding, but again toxic, and need to reapply a zinc-rich primer or paint locally after welding.

If site-welded from non-galvanized stock then painted it is unlikely that adequate surface preparation before painting will be done and painting will become an never-ending maintenance task for the owner.