To be fair, the app being posted is the server application which you can run locally. You do not need the phone application to use the web application.
This is unrelated to local first, just because you don't need to set up your own backend (not a server, because they are all servers) does not mean that the app works purely on the client.
That's even worse. The whole point of things like this are you do all your computing with your own computer(s). People in this thread are just saying that it would be even better if you could only use your phone/laptop etc and didn't have to also set up your own server.
Yes, that's what "Self hosted" means, you install it on your own server. Your choice if it's something "cloud" based and remote from you, or resides in your home.
Normally when talking about fitness tracking we are talking about embedded systems with very limited capacity both in terms of compute power and storage. You can use on-device storage to buffer tracking data, but any relevant and long term assessment and storage of the data has to be done off-site.
They are commenting about smaller devices such as a smart ring or a smart watch. Running a web server on such devices is the best way to deplete their battery.
This app is not about biometric fitness tracking as in smart watches, it is about manually tracking workout information. In that case, there is no need for a server to start with.
Huh, I totally don't get this conjecture... What do you consider fitness tracking? In my mind a basic fitness app is essentially a replacement for a journal.
I suppose an alternative design would be to be on-device storage first, and then have an optional sync to a server (or laptop/desktop).