Please review itinerary for day visiting Schilthorn and hiking

You may have the order wrong.
Grutschalp is on the way up via gondola from Lauterbrunnen. Or you can hike up.
The trains from Grutschalp go to Murren.
The gondola from Murren goes to Brig.
The gondola from Brig terminates at Schilthorn

First train up from Lauterbrunnen to Schilthorn via Murren (stopping at Birg for Thrill Walk) (1 hour) Might take a little longer. You take a gondola to Grutschalp, train Muerren, gondolas to Birg and Piz Gloria

Visit Schilthorn and have breakfast at Piz Gloria (2 hours) Won't take two hours unless you are pokey

Train down to Grutschalp (30 min) Gondolas to Brig and Muerren then train back to Grutschalp, probably close to an hour

Hike Grutschalp to Murren (1.5 hours) OK

Walk around Murren, have a snack (1 hour) OK

Hike Murren to Gimmelwald (30 min) OK

Train to Stechelberg (30 min) Gondola to Stechelberg

Hike Stechelberg to Lauterbrunnen with stops at Staubbach Falls and Trummelbach Falls (3 hours) Probably a good estimate

Thank you, I do understand all the steps to get from Lauterbrunnen up to Schilthorn, I just omitted them for brevity.
The goal was to get to Shilthorn first and then do each of our hikes, so wanted to make sure the time was sufficient and not overly ambitious. I really appreciate the input of this forum. It has been invaluable. Thank you!

The OP might have been thinking that the Schilthorn tends to cloud up later in the day, so do it first.

Back to the original question, yes. Doable, especially if you know the cable car schedules (and don't have to wait 30 min for the next one). Selenam, click on my name and in my profile you'll find a link to our pics and maps and schedules for the region.

Shoe, your page is fantastic! Thank you so much! This will be very helpful :)

You'll have already taken the train from Grutschalp to Murren; going back to hike it means three passes over the same stretch.

In the summer, you can descend to Birg, do the Thrill Walk, then hike down to Allmendhubel, stopping at Schilthornhutte for a snack. Continue walking or take the funicular to Murren and cable car to Gimmelwald. See https://schilthorn.ch/cmsfiles/SommerGuide_2018_d_e_f_web_1.pdf for the summer hiking trails.

I would straight to the top to Schilthorn first.
Stop at Birg for the Thrill Walk,
Instead of walking Grutschalp to Murren take the Funicular to Allmendhubel and take the North Face Trail, M2 on Shoe's map of Murren Hikes. It is beautiful! then walk down to Gimmelwald, then on to Stechelberg.

Just don't miss hiking down from Gimmelwald into Stechelberg - this was our favorite hike of our entire time there !

If you want to look around at all, or say hi to the goats or other wildlife on the way, or stop for photos along the way 30 minutes may be a tad quick.

It is however more of a walk than a hike - all of it is paved, except if you take the fairly steep shortcut which is all down with some steps.

The walk from Grutschalp to Muerren is very nearly level all the way.

I suggest checking the weather before heading up. All the hotels have a channel with cameras at the top, and there are screens at the cable car (gondola) stations.

Have you considered hiking up to the Schilthorn?? It is not an easy hike but it is incredible. It is about 10 miles round trip with about 5000 feet elevation gain. You can hike up one path and then come down another, both start in Murren.
It is an all day event and I don’t know if it would fit into your itinerary. PM me if you want more details.