Armoured shirt under a normal jacket versus a proper bike jacket, is the base layer thing a real substitute in city heat

You are giving up abrasion resistance, which is the half of the job the armour does not do. The garment standard has a class specifically for base layers worn as part of an outfit, and that class requires impact protectors but no abrasion performance at all. Armour handles the impact, the outer fabric handles the eight metres you slide afterwards, and a work shirt handles neither.

I own both and use both. Armoured shirt for 15 minute trips across town in summer where the alternative would honestly have been a t-shirt, mesh jacket with proper armour for anything longer or faster. It is a harm reduction tool, not an equivalent, and anyone telling you it is the same thing has not looked at what the labels claim.

The thing that changed my mind on city heat was that airflow beats fabric weight. A proper mesh jacket at 40 km/h moves more air across you than a linen shirt does, because it is designed to funnel it rather than flap. I stopped reaching for the armoured shirt entirely once I had a mesh jacket that fitted, and it cost about the same.

Check where the armour actually sits when you are on the bike, not standing in the shop. Mine rode up on the forearms in a tuck and left the elbow pads halfway to my wrist, which is a very expensive way to carry foam around. The good ones have a strap or a sleeve that holds position.

One more consideration nobody raises: if you are ever in an incident with another vehicle, what you were wearing tends to come up. Not legal advice and it varies enormously by country, so ask someone who knows your jurisdiction, but it is worth being aware that the choice is not purely about your own skin.