Do people actually use them like this? I'd assume its part of the joke, but then again I'm so old neither I nor my contacts use any emoji - we all "xD" and ":'D" each other like we did during the war I ought to tell you younglings about.
I think it's those who didn't grow up in the internet age and came here much later. What Gretchen McCulloch calles "Pre Internet People" in her book Because Internet. Meaning they grow up before internet and were introduced to it much later. Think of boomers who get their first smartphone because their son in law is afraid they get lost in the forest or something. You get the vibes.
So I think in this case it is part of the joke, making fun of older conservatives
I think "I'm so old" doesn't make the most sense as reasoning, considering that especially Boomers and old people are often famous for over-using emojis.
My (very kind, sweet, wholesome) 70 year old boss at work uses a lot of emoji. Not to this extent, but more than one a text seems like a lot to me and he has used 4-5.
I use :) and :D really much (the former extremely much, like really in every second sentence). :( also sometimes. But there are also quite some Emojis I often use, especially 😅. And being on Discord, there are also many custom emotes like :KEKW: I frequently use.