By "older update" I assume you mean an older version of iOS.
Check out https://en.wikipedia.org/wiki/List_of_iPhone_models#Supported
and (further down the same page) https://en.wikipedia.org/wiki/List_of_iPhone_models#Unsupported_(64-bit_CPU,_2013_to_2017_models)
It groups the different models by latest iOS they support. Now, my wife wanted a small smartphone, not some giant glass slab, and she did not need a ton of features, so she bought a refurbed iPhone SE (1st gen) from BestBuy. The latest iOS it can support is iOS 15.8.3, so it's never going to end up with anything newer. She's perfectly happy with it.
My point is that if you know which iOS your son wants, you can look it up and see which models have that as it's most recent version it can support. Then, whatever iOS the phone currently has when you buy it, Apple cannot force it to update to any more recent version. Also, for any given model, you cannot arbitrarily choose to downgrade it to an older version, even if it's the version it originally shipped with.
I recently decided to migrate from an Android phone to an iPhone. I chose an iPhone SE 2nd generation (also relatively compact), which shipped with iOS 13.4, but it updated to the most recent iOS 17.6.1, and (reportedly) is supposed to support iOS 18 when it comes out, and I believe it was the oldest/cheapest/smallest model that will do that. It's my Goldilocks phone. Actually, it's the second "renewed" one I got from Amazon. The first one turned out to have a defective digitizer, which resulted in a dead 7mm strip down the right side that would not respond to touch, so Amazon authorized a full refund on return. The second one seems fine so far, although I though I had "bricked" it with the update to 17.6.1. It got better on the 6th attempt to restore it.
My point with that ramble was to not be afraid to buy a refurbed older model as long as it's from a reputable vendor.
[edit] Oh. I see the problem now. The iPhone 11 shipped with iOS 13, and I'm assuming that's the iOS he wants. There doesn't seem to be any models whose most recent iOS will be higher than 12.5.7 and lower than 15.8.3. The only solution I can see is to decide which version(s) of iOS outside of that gap he would find acceptable, and then get an iPhone that maxes out at that version.