Endless crash reports

Something is still wrong with your home folder permissions. Post the output of the shell command below:

ls -Odel Library

Back up all data now. Before proceeding, you must be sure you can restore your data to its present state

Step 1

Launch the Terminal application.

Drag or copy — do not type — the following line into the Terminal window, then press return:

chmod -R -N ~

The command will take a noticeable amount of time to run. When a new line ending in a dollar sign ($) appears below what you entered, it’s done. You may see a few error messages about an “invalid argument” while the command is running. You can ignore those. If you get an error message with the words “Permission denied,” enter this:

sudo !!

You'll be prompted for your login password, which won't be displayed when you type it.

Step 2

Boot into Recovery by holding down the key combination command-R at startup. Release the keys when you see a gray screen with a spinning dial.

When the OS X Utilities screen appears, select Utilities ▹ Terminal from the menu bar. A text window opens.

In the Terminal window, type this:

resetpassword

That's one word with no spaces. Then press return. A Reset Password window opens. You’re not going to reset a password.

Select your boot volume if not already selected.

Select your username from the menu labeled Select the user account if not already selected.

Under Reset Home Directory Permissions and ACLs, click the Reset button.

Select  ▹ Restart from the menu bar.

I was, of course, hoping to just use Migration Assistant to transfer everything over.

Don't use Migration Assistant. Use Setup Assistant. It launches automatically when you first boot the new Mac.

...if I did nothing more would the problem I am having get transferred to the new machine?

Yes.

Figure out how such a mess got created in the first place. That's more than I can do.

I was having a nearly identical issue as the OP. 10.11.3 El Capitan, MacBook Air 5,2, every Apple app I used crashed at some point (TextEdit, Notes, Safari, Mail, Pages, Numbers, Message, FaceTime, iTunes, App Store), and sometimes several times in a row. I got as far as

chmod -R u+rwX ~

Unfortunately I have to retract my statement that following these steps helped me, too. I thought they did, as my computer ran better and nothing froze all day. But the next day things were back to their old ways.