There are other countries which have something like this. It changes the political culture, but not completely. And there are problems:
How many candidates can run and get money?
You seem to assume that it is just the primary winners of the big two parties. How about the primary winner of a 3rd, or 4th, or 100th party? Do they all get the same sum?
- One option is to tie campaign funding to the outcome of the previous election. But who administers the money? The previous candidate? The party? The new candidate? And won't that give an unfair advantage to the big parties? Or a fair advantage?
- Another option is to require supporting signatures before a candidate gets onto the ballot, and to give funding only to those with a significant number of signatures. But how many?
How to fund the primary?
Money is required early on. If there is public funding for the main election, the overall tendency to donate for campaigns could go down, which makes candidates even more dependent on the remaining donors.
Can candidates and parties spend more?
There is an assumption that spending more brings more votes. So candidates would be tempted to ask for campaign donations anyway. The scheme would merely increase the total amount of money, it wouldn't slow the spending race.
- Preventing candidates and parties from spending more runs into freedom-of-speech problems. If a group of political activists really dislike one candidate, surely they can buy ads against him? Any collusion with the campaign of the other candidate would be for the investigation afterward.
- Can a candidate save leftover funds from the last election for the next one?