I have a CL started on this, but priorities shifted.  I hope to come back to it in a couple weeks.  Please let me know if someone would like to take this over.

Could you post a link to the CL? I might have some time to work on this while waiting on @bettes to address  crbug.com/600468 .

As discussed with @dschuyler, I'll take over adding the "edit" functionality and moving both "edit" and "delete" under a popup menu. Potentially converting the URL list to use iron-list will be decided separately based on performance observations on long URL lists.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/4c190f9618256c9175592abf6343834d96badb4e

commit 4c190f9618256c9175592abf6343834d96badb4e
Author: dpapad <dpapad@chromium.org>
Date: Sat Apr 09 01:41:41 2016

MD Settings: On Startup, extract dialog to its own file.

Also:
 - Wrapping the dialog with dom-if such that it only exists in the DOM while open.
 - Removing unnecessary usage of PromiseResolver.
 - Removing unused "prefs" property from <settings-startup-urls-page>.
 - Simplifying validation logic.
 - Adding missing tests.

This is in preparation of adding "edit startup page" functionality to the same
dialog.

BUG= 601514 

Review URL: https://codereview.chromium.org/1869883003

Cr-Commit-Position: refs/heads/master@{#386269}

[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/on_startup_page.html
[add] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/startup_url_dialog.html
[add] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/startup_url_dialog.js
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/startup_urls_page.html
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/startup_urls_page.js
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/on_startup_page/startup_urls_page_browser_proxy.js
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/resources/settings/settings_resources.grd
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/browser/ui/webui/settings/settings_startup_pages_handler.cc
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/test/data/webui/settings/cr_settings_browsertest.js
[modify] https://crrev.com/4c190f9618256c9175592abf6343834d96badb4e/chrome/test/data/webui/settings/startup_urls_page_test.js

Status update. I have the "edit" functionality working locally. The changes are broken down as follows,

1) Adding C++ support for editing, https://codereview.chromium.org/1877633002.
2) Adding dots popup menu and moving "Remove" option there, https://codereview.chromium.org/1876283002.
3) Hooking up edit functionality, https://codereview.chromium.org/1882483002 (adding tests still WIP).

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/413b72afce529a79146d60ad410a8600f8cd3098

commit 413b72afce529a79146d60ad410a8600f8cd3098
Author: dpapad <dpapad@chromium.org>
Date: Tue Apr 12 19:40:00 2016

MD Settings: Adding C++ support for editing startup URLs.

BUG= 601514 

Review URL: https://codereview.chromium.org/1877633002

Cr-Commit-Position: refs/heads/master@{#386767}

[modify] https://crrev.com/413b72afce529a79146d60ad410a8600f8cd3098/chrome/browser/resources/settings/on_startup_page/startup_urls_page.js
[modify] https://crrev.com/413b72afce529a79146d60ad410a8600f8cd3098/chrome/browser/resources/settings/on_startup_page/startup_urls_page_browser_proxy.js
[modify] https://crrev.com/413b72afce529a79146d60ad410a8600f8cd3098/chrome/browser/ui/webui/settings/settings_startup_pages_handler.cc
[modify] https://crrev.com/413b72afce529a79146d60ad410a8600f8cd3098/chrome/browser/ui/webui/settings/settings_startup_pages_handler.h

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/9035bfdadfeece36231732a0dd21fbd0ae200730

commit 9035bfdadfeece36231732a0dd21fbd0ae200730
Author: dpapad <dpapad@chromium.org>
Date: Wed Apr 13 23:53:09 2016

MD Settings: On Startup, moving remove URL option under a dots menu.

 - Extracting new element <settings-startup-url-entry>
 - Implementing dots menu, that currently only has the "Remove" option.
 - Adding test.

This is in preparation of adding the "Edit" option under the same menu.

BUG= 601514 

Review URL: https://codereview.chromium.org/1876283002

Cr-Commit-Position: refs/heads/master@{#387158}

[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/app/settings_strings.grdp
[add] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/on_startup_page/startup_url_entry.html
[add] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/on_startup_page/startup_url_entry.js
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/on_startup_page/startup_urls_page.html
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/on_startup_page/startup_urls_page.js
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/on_startup_page/startup_urls_page_browser_proxy.js
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/resources/settings/settings_resources.grd
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/browser/ui/webui/settings/md_settings_localized_strings_provider.cc
[modify] https://crrev.com/9035bfdadfeece36231732a0dd21fbd0ae200730/chrome/test/data/webui/settings/startup_urls_page_test.js

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329

commit 8b14059a8b62eac0b7ed74b2fc233aecb8cf8329
Author: dpapad <dpapad@chromium.org>
Date: Thu Apr 14 02:55:24 2016

MD Settings: OnStartup, implementing "Edit" functionality.

 - Modifying <settings-startup-url-dialog> to act both as "Add" and "Edit".
 - Bringing up dialog when user taps the "Edit" option.
 - Adding tests.

BUG= 601514 

Review URL: https://codereview.chromium.org/1882483002

Cr-Commit-Position: refs/heads/master@{#387218}

[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/app/settings_strings.grdp
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/compiled_resources2.gyp
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_url_dialog.html
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_url_dialog.js
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_url_entry.html
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_url_entry.js
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_urls_page.html
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/resources/settings/on_startup_page/startup_urls_page.js
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/browser/ui/webui/settings/md_settings_localized_strings_provider.cc
[modify] https://crrev.com/8b14059a8b62eac0b7ed74b2fc233aecb8cf8329/chrome/test/data/webui/settings/startup_urls_page_test.js