Long post ahead...
You, like the entire GaryJohnsonJillStein-Internet, apparently do not realize that "federal funding" for political parties is basically zero lol. It's financed by a box you can check on your tax forms where you have the option to donate $3 to the fund, not even assigned to any particular party. This in recent years has been around 10 million (imo silly) tax payers, aka $30 mil a year. This gets distributed every 4 years for presidential races. So $120 million is available. In reality it's higher but I'll get to that in a bit.
Who gets it and how? Well, the simplest answer is that in the 2012 election almost all of what was given out ended up going to the primaries or the national conventions, not to any particular general election candidate (though funding for national conventions was removed from the law in 2014). And only like $10-20 million could even go to a convention, since that amount is fixed in the law and tied to inflation. So why the heck was "the majority" of $120 million $10-20 million? Well, if an individual candidate accepts any of the money, then they are required to follow a whole bunch of campaign spending restrictions. So in reality no Democrat or Republican took it in 2012. McCain took it in 2008, like everyone had before him, and Obama absolutely thrashed him on donations by not taking it and being subject to the fundraising caps. So in 2012 again Obama didn't take it and Romney didn't either. This results in money compounding over time in the fund. Today there's more like $300 million in it, not $120 million.
So could Gary Johnson, or whoever the next libertarian nominee is, qualify for $300 million in free federal money? Well, no. A chunk of that isn't even up for grabs for the general election because it is reserved for primaries. It's not entirely clear how much they would be eligible for, but it's not the entire amount. And in any case, of whatever the base is, they would only be eligible for a percentage, proportional to the vote they got. In 2000, based on the Reform Party getting 8% of the vote in 1996 (Perot's second time, and close to what Johnson will get), the Reform Party qualified for $12.8 million for their nominee. Based on the 19% in 1992, they had qualified for $29 million in 1996. Both of those totals are certainly more than Gary Johnson has raised either of the past 2 times and it would make a difference, but only a marginal one. $12.8 or even $29 million doesn't buy you squat in TV ad time. Even if they actually do qualify for all $300 million, they would still only get $24 million. That's a ton more than they currently raise, but it's still nothing compared to the major parties. Still not much TV time. Clinton has ALREADY spent more than twice that much on TV ads, and it's not even September yet.
The third parties get support because of "true believers" in their message or because of supreme distaste with the other two major nominees (not likely to happen quite the same way as in 2016 again lol). More money, until it actually matches what major parties can do, is not going to move the needle. It's not as though $1 million of contribution = 8% of votes means that $60 million of contribution means 480% of votes. There are very, very diminishing returns on money, as you can see from Clinton v Trump. She's outraised him by way more than she actually leads him in polls.
The Reform Party went from 19% of the vote to 8% to 0.4%. I recognize that is largely because they nominated a bigoted idiot in 2000, rather than someone sensible like their founder Ross Perot. But wow that $12.8 million of federal funds they had for the 2000 cycle really made a world of difference didn't it! And so did that $29 million they had for 1996.... when they nominated literally the same guy as last time and got less than half as many votes despite all that extra cash.
Look how successful all these campaigns have been! List of candidates who took federal funds by year (both primary and general):
2016: Martin O'Malley, Jill Stein
2012: Buddy Roemer, Gary Johnson, Jill Stein
2008: Joe Biden, Chris Dodd, John Edwards, Mike Gravel, Dennis Kucinich, Ralph Nader, Duncan Hunter, Tom Tancredo, and finally the aforementioned John McCain
.... do I need to go on?
It's just not a good deal to take it if you have any actual shot at winning. The system was designed decades ago when PAC spending amounted to like 1% of election spending, and it no longer makes any sense in our money-filled politics. Having how much you can independently fundraise capped is too hard of a pill to swallow.