Generally means in most cases, or speaking broadly. Generally speaking, generally available, it generally works.
It hedges, and that is its job — generally admits exceptions without listing them. Generally speaking is a set phrase that says the same thing twice, which is why editors cut it. In general is the phrase to use at the start of a sentence.
1adverbgenerallyIn most cases; speaking broadly.
“I generally walk to work, but I take the bus when it rains.”
generally is a hedge.
General plus -ly, from Latin generalis, of a whole class, from genus, a kind.
Middle English
generally
English, today
generally
From Middle English generally, generalliche, equivalent to general + -ly.
Etymology from Wiktionary, CC BY-SA