Gmail makes things more secure, but adds a twist when they don't explain how to add aliases to Gmail Accounts in their help files. New method was just tested today (Aug 22, 2016), and found to work. It may not work later if they change things again.
1. If 2 step authentication is not enabled in your account, follow these instructions.
To enable 2-Step Authentication on your Google/Gmail account
Go to https://myaccount.google.com/security
Log in using your email and password for your gmail account
Under Password & Sign-in Method, click the arrow beside "Off" to turn it on.
Enter your password and prompt for verification method (Text message or phone call)
Enter the code where prompted then click "TURN ON" go to top to continue with Alias instructions.
2. Go to https://security.google.com/settings/security/apppasswords
3. Click "Select App" - Select "MAIL" then "OTHER" and enter a custom name IE "Gmail1". A 16 digit generated app password (Keep this open as you'll need to copy/paste later)
4. In a new window, open GMail, click Gear Icon, and then Settings then Account and Import
5. Send mail as - click Add another email address you own"
6. A popup window will appear, enter Your Name, and the "Alias" email you wish to use. Uncheck "TREAT AS AN ALIAS" then NEXT STEP
7. Change SMTP server to - smtp.gmail.com, Change Username to your gmail address, and then copy/paste the 16 digit generated app password.
8. You'll be prompted with verification options, which I chose to enter a code emailed to the alias email address that I entered.
Here are the step by step instructions with example images
Setup Gmail to authenticate to, and use, their own SMTP servers for sending messages from those non-gmail addresses.
Assuming you already have 2-Step Authentication enabled for your Gmail/Google account, go to the Accounts and Import tab in Gmail Settings, and Add another email address you own, enter your name and email address, and uncheck Treat as an alias.
Then click the [Next Step] button.
For the Password, you'll have to create and provide an App Password generated in Google Accounts App Passwords page at https://security.google.com/settings/security/apppasswords.
Then click the [Add Account] button.
Creating an App Password
Creating the App Password is relatively easy. Click through to the Google Accounts App Passwords page using the link above. Then select the type of App password you're creating -- use Mail -- and choose Other (Custom name) as the device. Give your app a name, such as "myGmailSMTP", then click the [Generate] button.
Be sure to use the password immediately -- as of this writing, once you click [Done] on theGenerated app password window, there is no way to retrieve or view this password again. You can use the same App Password for several "Send mail as" accounts used in this gmail account, or you can simply create new ones as needed -- I suspect that Google's intent is that they be used only once for each (send mail as) app.
You can also edit and set your current "Send mail as" accounts to use this process, too.