No. Think of how a fan shroud works. First, the heat is not trapped. This is what the thermostat and fan are for when you’re not moving and using convective cooling. The shroud acts to direct the airflow past the engine and trans, the belly pan contains it until it reaches the back of the motor where it can exit.
Of course we’re going to do as each decides is best. The idea of a discussion is to exchange ideas and perhaps we come away with tidbits of information we might not normally have had or considered. If you consider examining all possible contributions over thinking, then that is definitely what I am doing- all the time. It’s in my nature. That’s why I went into engineering in the first place 
I suppose… I’m still not certain those belly pans don’t actually hold in more radiant heat and block some airflow from underneath. I dunno. Kind of like a hood. I’d imagine an engine would run slightly cooler without one. But I’m not certain. I think I’ll take the hood off my Lesabre and find out. It’ll look kicka**. 
I think modern cars have problem not with heat dissipation, but rather with heat retention.
I’m having a “project” with 14-years old Prius for my younger daughter and this thing has a special “heat accumulator” where it sucks the hot coolant from the engine every time you shut it down, so on the next start it would be able to get warm again faster and get to the best thermal efficiency.
My 2019 HJonda Accord Hybrid has a special heat exchanger in the exhaust system to get coolant/engine warmed up faster, as it plain produces too little heat just from idling.
I might assume “belly panel” would help here too, together with a pad on the hood panel, which was there for decades now
Heat dissipation is probably better for the engine. Heat retention is probably better for emissions. My guess, anyway.
Who knows. If anyone’s car overheats (or underheats) by removing a belly pan, maybe they’ll report it.
Be sure to flip over the air cleaner lid too!
Did all that stuff back in the day…
Yeah, it seems counter-intuitive. But recall, one of the primary goals today is to reach operating temp as quickly as possible and yet prevent overheating. By defining the airflow path, it helps to do both. The mechanical guys just shake their heads when everyone else questions why we need this baffle, holes or slots in this spot etc. Seems everyone is a mechanical engineer 
Well, I can’t “flip the breather”. I guess I could install a cold air intake with the filter sticking up in the wind. I wonder if people would take pictures of my car at the ole Wal Mart.