Why is the following not correct?
There was a lot of public at the concert.
Public, under most circumstances, is an adjective. The grammar you have used would indicate that it was a mass noun, which it is not.
The public can be thought of as a set phrase that means "people in general, not part of some group determined by context". So, if the people running a museum talk about "the public", it means people who don't work for the museum - and will generally be used about members of the public who come to the museum. It's also usually used for generalisations, and people will usually accept that there are exceptions without needing to say so.
"The public won't know why they shouldn't touch the exhibits. Thousand-year-old artefacts being fragile just isn't something that goes through their minds."
The public also behaves as a grammatical plural most of the time, despite looking like a singular:
The public are easily manipulated.
(It's possible that this could be ellipsis of "members of the public")
Because it acts like a plural made up of distinct, countable individuals, I would usually use many to indicate large quantity:
There were many of the public at the concert.
That doesn't come across as wrong, to me, just unnatural. In that case, and I can't really explain why, I would use:
There were many members of the public at the concert.
Though I'm not sure why you can just say many people, and then specify in the audience to make sure people know what you're talking about. Also, despite some people's insistence on rules about countability, it would be common to say "lots of people", so you could have>
There were lots of people in the audience at the concernt.
Some hidebound teachers might insist on it being "many" rather than "lots of", but it would be a rare native speaker who would bat an eyelid.
Important point, though - if public is being a noun rather than an adjective, it needs a definite article - the public.