I don't rely on any website for recording my mileage. I don't know the likelihood of any given site shutting down but it seems your records could be at risk if the one you relied on suddenly wasn't there.
So I keep mine on my desktop computer at home. Once upon a time I logged miles in a spreadsheet (back in the 90's and it started with running). When that got too unwieldy, I migrated that to MS Access. By then I think I'd incorporated cycling miles but eventually I hit some kind of limit when the entries exceeded 9,999. This led me to MySQL, which I have been using ever since, with the added benefit that it is free and independent of the operating system.
Using html as a front end and to display results is also free, as is PHP which I use for logic and Apache which is the web server.
My application has evolved over the years to be a workouts database, not just a means of logging miles. I track all the kinds of exercise I do with it, from push ups to hikes to bicycle touring or commuting, etc. It will tell me how many miles I've run in a given pair of shoes and even the breakdown of the types of surfaces I ran the miles on and it will tell me what bicycle I rode the miles on.
A couple of years ago, as an unemployed software developer with time on my hands, I found that I could import data from Google's Mytracks and from my Garmin watch so that I did not have to enter any distance or time information for a ride, run or walk. And it would tell me where I'd been during the workout.
My latest enhancement is to create a table of bicycle parts and one of installed parts and relate them to the rides I record. So now, like with shoes for running or hiking, I can tell how many miles I have on bicycle parts. I have 259.2 miles on newest tire, a Schwalbe Marathon Plus, in case you're wondering.
Admittedly, it is a bit more hassle when I get a new computer as I have to go through the installation and configuration of Apache server, MySQL, etc. But I don't lose the data due to it and I never have to fear that because of the disappearance of some website I have no control over.