I am struggling to see why this would imply the first person singular being the subject of this sentence. There is only an adjective, infinitive and a genitive serving as an object. Does this kind of structure always imply first person or can it be other things to depending on context?
It is a set phrase. If you mean a different person you have to say it but if you want to say "I am happy that..." or "I am sure that...", рад (рада) and уверен (уверена) are enough. Clichés work a bit different from sentences you make up freely.
Very clear thank you, when you first encounter something new, especially in a new grammatical context it's difficult to assess whether it is idiomatic or not