Earlier quoted context omitted.
Which third parties do you mean?
My understanding of lightning is, that unless you run your own node (which requires a lot of funds and needs to be online 24/7), you are using a third-party node for these payments. The positive would be, that there are (I presume, don’t know) far more nodes/services available to choose from than there are credit-card firms. This is just what I remember from reading into lightning a few months ago.
As far as I understand it: When a channel you have open is closed by the other side of the channel your wallet needs to know this within a certain time (24 hours? Can't remember).
So to avoid having to read the blockchain updates once a day, you would subscribe to some service that tells you when one of your channels is closed.
I also think you only have to do that if you distrust the other side of the channel. If you select a channel partner (a friend, a well known institution, a bank?) and only keep a small amount in that channel, I think you can just trust them and not sign up for any "watchtower" to watch if they try to cheat on you.
PS: Looking up the growth of the Bitcoin blockhain it seems that even if you do the watching yourself, that would only mean to read less than 1MB per day? Here is the data: