Concrete – Running irrigation pipes under a large concrete pad

I am not a mason, but my guess is this will be difficult to do it well. My advice is even more suspect since I cannot see the stairs themselves and the extent of the problem.

A thin veneer over some faces and a thick one in other places will chip and crack, especially if water ever gets in and freezes. But the major problem is the existing cracks. Those cracks may come from settling/frost heaves, etc. That will leave this new layer prone to the same problem. In order to prevent this, the new layer would need to be reinforced to strengthen it, and it would need to be thick enough to be able to resist cracking when one half of the stairs tries to move.

Do these stairs rest on bare ground? Or do they have a proper foundation underneath to avoid frost heaves? My guess is the cracking suggests they are moving around with the weather.

A poor fix that fails in a couple of years is worse than doing it right the first time.

The best solution probably is to remove them and put new stairs in.

In most US jurisdictions, from a legal perspective, the work in the basement would fall under the local building code and its permitting, licensing, and inspection requirements...e.g. a splinkler contractor could not run the pipes inside the house and a plumber would be required, building department fees would be applicable, etc.

The pipe running through the basement that conceptually would be part of the irrigation system, is both legally and from a life safety standpoint part of the domestic potable water supply [it may be possible to get the building department to treat it as a domestic non-potable system, but that's less likely than a flat "no" from the building department]. Even if you get it approved, you'd still want a vacuum break where the pipe exits the building and there's a risk of contamination.

Anyway from a practical standpoint, one thing you're facing is the standard practices of the irrigation installer tradebase. They own Ditchwitches and shovels. They don't usually dress for walking across homeowner carpets and aren't in the habit of pulling plumbing permits. This is pushing things outside their business model...and reducing their revenue. Maybe you see a portion of whatever savings the contractor realizes, maybe the contractor just passes on the job [it depends on how much work they have, and you want the person who is busy].