Actually, as another kind of important followup to this: in general the OS-provided controls are already accessible and extending them will get you into trouble.  So if it's a button and it already does the button things, everyone is going to already know how to pull button things from it without you doing anything, and you trying to make it special is probably going to end frustratingly.

I just tested with the following code:
    public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();
            var b = new Button();
            b.Text = "Press me";
            this.Controls.Add(b);
            var b2 = new Button();
            b2.Text = "Press me 2";
            b2.AccessibleName = "B2's accessible name";
            this.Controls.Add(b2);
        }
    }

I tested by:
Creating a .NET Framework 4 client profile WinForms app in VS 2010.
Creating a .NET Framework 4.8  WinForms app in VS 2019.
Creating a .NET Core 3.0 WinForms app in VS 2019.

I tested with both NVDA and JAWS.
When tabbing between the buttons, all scenarios spoke exactly the same.
They said "Press me button" for the first button, and "B2's accessible name button" for the second button.
JAWS also spoke a help tip telling meI could press spacebar to activate the button, but I'm sure that can be turned off in JAWS settings.
There was no duplicate speaking of text and accessible name that I noticed, and the buttons always spoke.

I am running Windows 7, so perhaps running on Windows 10 gives different results.
Or perhaps you are using a different framework version than I tested with.
If you like, you can email me your program/solution at [email protected] and I will test it to see if it works any better for me, or perhaps I can spot the difference that is causing the issue.

Quentin C is kind enough to publish the playroom statistics:
https://qcsalon.net/en/stats?general
Under the "Screen readers used" heading you can see that JAWS is used by 43.84%, and NVDA is used by 51.5%, out of 13,244 users.
That accounts for 95.34% of the users included in the stat.
Personally, I would only worry about supporting NVDA and JAWS on Windows.
And as Camlorn said, using the operating system controls (which WinForms does) should not require any extra work on your part to get decent accessibility.
Speaking extra notifications through Tolk can be a good idea if used appropriately.

Ian really is a wonderful gentleman. He always gives very helpful answers without ever being snide or snarky, neither contentious nor contrary.

Kai