He's full of crap.
A 440 is the standard whether you're playing at home or Carnegie hall (well ..... Carnegie hall might request A 442 for some players)
He probably doesn't want to deal with it.
Thing is ....... that's
just enough of a pullup to make it a bit of a pain. It's hard to justify asking the customer to pay for a second tuning when it's that close and many tuners aren't that good at salesmanship and don't like to have to spend maybe an hour explaining to the person why it'll quickly go out of tune again ....... but it's enough that if you just tune it ...... it'll go out quickly enough that they'll be calling you back because "your tuning didn't hold"
So the only thing you can do if you want your customer to be happy and recommend you to others etc. etc. is to take an extra chunk of time and get it up and stable or you risk having them tell people that you're a bad tuner.
The reason that pianos go out of tune when you pull them up (or down if they're high ..... that's even harder) isn't string stretch. It's because you add tension to the piano. For example ...... if I were to pull up a piano that's 1/2 step low (much more than yours) I'd add about 6,000 lbs of tension to it. That's enough, iron plate or not, to compress the piano thus detuning it.
Even worse ..... it starts as you go along so the very first strings you tune are affected the most and the last ones ..... almost none.
Plus .... shorter strings ...(the high C is only a couple inches long) are affected very diiferently than a big ol' several feet long bass string.
Personally, I would tell a customer in your situation that it's not gonna hold very well this time and it should be tuned again withing 3 months. I'd carefully explain the process (even though lengthy explanations take time=money) and then they'd do whatever they did. Some would call in 3 months and have it done ...... some would make the appointment right then and I'd just show up in 3 months, ..... many wouldn't get it done thinking it sounded fine as it was. VERY many people can't be convinced to tune their pianos even yearly.
But I had over 3,000 customers and can only do maybe about 1,000 a year so I basically did my job well, took time to explain and then didn't think of it after that. Whether they tuned it or not I was overloaded anyway.
So for whatever reason he doesn't want to deal with it but he's absolutely wrong about A420 although that's a fairly creative excuse.
But the fact that he told you that means either he doesn't know what he's talking about ( plenty of crappy tuners out there that spout off nonsense) or he's lying to avoid pulling it up.
In either case, that'd make me want to find a different tuner.
Where in Fl are you?
Now ..... there are times when I will tell a customer that there's no need for them to have their piano at A440. If it's 1/2 step low but it's a little old lady that ONLY plays hymns all by herself for fun and will never ever play along with any other instrument ever ...... then why should she pay extra dollars to have it standard? It'll be just fine as long as it's in tune with itself. And despite some peoples' claims ...... if I do a really good job of tuning a piano to itself, 99% of the people on this board would play it and have no clue that it was 1/2 step low unless they tried to play with someone else.
Also, occasionally I'll see a piano where the strings are so rusty that pulling it up dramatically increases the chance of string breakage. And at $20 a pop .... a few strings isn't too bad ..... but 50 of them adds up!
But anytime a piano is gonna be played with anything else ..... it has to be at standard pitch.