I've been a long-time user of Twitter (user #82,903 joining in December 2006) and while I've had ups and downs with the service and generally think of it as a guilty pleasure I continue to use it. However, I have always thought of Twitter as ephemeral. Tweets to me are very different than blog posts. They often do not stand on their own but are really meant to be understood in the context of the moment.
Many years ago, before the Twitter API introduced OAuth requirements I used to run a script on one of my machines that would delete tweets after a while. This became annoying to manage after the API changed and I stopped running it.
We never store any of your tweets on our servers, we only delete them on your behalf.
How terrible would it be if a service like this actually archived the tweets and then sold it for advertising purposes to 3rd parties. Terrible, and in the modern Internet it would be completely unsurprising so I was glad to see this stated clearly in their very human readable terms of service.
But what about archiving? While I do view my Tweets as ephemeral I do keep them for myself. I see little to no value for others in what I tweeted 10 years ago, but I do like to have a diary like record for myself. That is where Pinboard comes in! Pinboard has always had a feature that archives the tweets of up to 3 accounts and I've had that on for several years so I still have a personal archive of everything which I like. By the way, it's an archive I pay for and know isn't shared with others. I love the ethos of Pinboard!
I would encourage you to ask if there is a reason you want your old tweets around and if not have Tweet Delete take care of it for you. Think of it as helping poor Twitter not have to keep so much data around.