Verizon sells one they claim is LTE capable. Consumer Cellular has a couple of grandpa-friendly models. Or you could just get something like an Apple Watch 3 with LTE, which achieves a similar goal.
I haven't used that phone in 6 months or so because I switched to an Apple Watch series 3. I've never had a smartphone as my actual phone. I synced it to my SIM on my wife's phone with my Apple ID, then reset her phone back to her stuff.
The watch works great as my only phone. Eats batter power, about 15 hours per charge, less if I talk or listen to music more than a couple hours per day. Looking forward to watchOS 5 because it will the phone self-manage wifi settings, which should fix a lot of the battery issues. Right now I can only connect to one wifi network (since I don't have a phone to sync to that manages it). Once I can connect to my home and office wifi, battery life will improve.
Not having a phone in my pocket ever again is a great feeling. Just the default apps on it. Wish I could stream podcasts. I still keep the dumbphone around the house in case I need to take an SMS (Apple Watch apparently doesn't handle those either, just Apple Messages).
My S1 definitely does text messages, not just iMessages, but like I said, it is tethered. If the LTE model cannot do standalone texts, that is a bit unfortunate -- I do get normal text messages periodically and I would hate to give those up entirely.
Although I don't know if it really fits the bill. You still need to pay for data access, it is far more expensive, and no private calls without some bluetooth headphones.
Essentially everything you'd want from a dumbphone, small screen, no real apps, basic phone functions and maybe email.
I am actually thinking I might pick up an LTE apple watch in a couple months when the next generation comes out, exactly for the reason of leaving my iphone at home pretty often. Less distraction, less willpower required to stay off it.