not a scientist, but this is my understanding of things too.

the "big bang" was originally coined as a pejorative way to describe the idea. it just stuck because it's catchy.

our physics goes so far back, to a certain point. and it looks like the math would go back to a zero size point. which is how we get to fractions of a second after the big bang. but that's just using similar math to: a plant is growing 1cm a day, it's 10cm now, so 10 days ago it must have been 0 size.

our physics doesn't describe the early universe beyond a certain point, so we can't actually tell what it was like. and we know there's 4 big problems in current physics, all of which would affect this:

• quantum mechanics and Einstein don't align
• quantum mechanics does not describe what collapses the wave function
• we don't know what dark matter is
• we don't know what dark energy is

and these are just the "known unknowns".

however, "the big bang" remains the most plausible model, describing the cosmic microwave background and the current expansion and distribution of matter within the universe.

some of the big things we know we don't know about it:

• what happened at t=0 and just after, if t=0 really was the start of everything or just some inflection point
• why we appear to have mainly matter and not matter + antimatter

I'm sure there's a bunch more things we don't know. but your characterisation seems accurate to me.