Outlook 2010 can no longer open Gmail

you need to do the following before you can setup gmail in outlook.

Change account access for less secure apps

To help keep Google Accounts through work, school, or other groups more secure, we block some less secure apps from using them. If you have this kind of account, you’ll see a "Password incorrect" error when trying to sign in. If so, you have two options:

Option 1: Install a more secure app that uses stronger security measures. All Google products, like Gmail, use the latest security measures.
Option 2: Change your settings to allow less secure apps into your account. We don't recommend this option because it can make it easier for someone to break into your account. If you want to allow access anyway, follow these steps:
Go to the "Less secure apps" section of my Account.
Turn on Allow less secure apps. (Note: If your administrator has locked less secure app account access, this setting is hidden.)

https://support.google.com/accounts/answer/6010255

Thanks, Sunil. I cannot get to the "Allow less secure apps" option. I am under the impression that I am the administrator. This is a home computer with two desktops and wi-fi access. I do have the 2-step verification and have had it since Oct 16, 2016. I tried using an app password but that did not seem to work. I have noticed that "Amazon" has "Has access to Gmail, Google Calendar, Google Contacts, basic account info". I have tried turning that off and it caused another issue but I do not recall what that was. I know I am not real enthused about Amazon having access. I think it is the same on the Kindle Fire that I use.

Two-step verification stopped working when you did the updates, because you were logged out.

If you want to keep 2-step verification, you need to go into your account, security settings, and generate an application specific password for Outlook.  Once generated, you use this password to have Outlook log into Gmail.

You can have 2-factor authentication (very important) and still use Outlook.

Thanks, Allen. I understand why this happened now. Since it appears there will be some more updates coming soon, I will be aware of this issue. I did go ahead and restart the 2-step verification, though I was unsure how this would affect the Kindle I also use. After one false start, I followed through and Outlook still works on my desktop and my Kindle appears to be OK. If I should have an issue with these, at least I know how to back off. You are correct in saying I want the 2-step, especially on the Kindle.
It appears that my link did not show up. Here is another attempt.
Link to partial solution.

Once I had Outlook working, I applied a 2-step verification and then generated an app password.

Glad my info was helpful.  You can also print a list of 10 emergency, one-time, passwords in case you ever get prompted for the 2FA code and don't have your phone with you.  I keep mine on a card in my wallet without any info on what they are for.

It restored my access to Gmail.