Have not stayed at E-Suites but friends have. As shown, it's a hike. Thoughts:
1) Can walk to the Toy Story Lot and take the parking shuttle, which looks to be 2/5 of the walking distance (it's right at the "31 Min" sign in the screen shot above). They have new security screening at this lot, which will allow you to bypass the security at the main Harbor Blvd entrance.
2) E-Suites South has a complimentary shuttle, unless something has changed. This will take you to Harbor Blvd security, likely the busiest. This HAS been reconfigured recently to prepare for Star Wars, so it's hopefully much improved. But we haven't seen the "crowds" yet.
3) As mentioned, consider an Uber/Lyft to the Downtown Disney entrance near Disneyland Hotel. This should be the least-crowded tents. If low-stress is worth a few bucks to you, this is a really nice option. And the walk through DTD is really nice in the morning. Mobile order from "Starbucks West" during your ride and should be ready for pickup.
I would probably try a mix of #1 and #3 myself. If you sense crowds are not bad, #2 would work. But I would call the hotel lobby and ask if their shuttle makes stops at other hotels. If it does, the loading/unloading can be frustrating in the morning.
Last thing: If you have strollers, make sure you are up to speed on the new rules before leaving hotel. Have fun