I have recently heard an idea from a local journalist that even surprised the moderator (greatly simplified):
Pay some amount for people that do NOT vote
Heuristically this should lead to better results because those interesting in taking the money are less interested in politics, more egocentric, thus the result will be reflect options from those who understand the value of the vote, value more the politics than immediate financial gain.
In point two, you claim that this system would encourage votes from people who "understand the value of the vote", but is that really true? Votes are very important in aggregate but a single vote, not so much - the overwhelmingly most likely outcomes of adding one vote are either 1) your preferred candidate wins by one more vote than they already would have, or 2) your preferred candidate loses by one fewer vote than they already would have. It is extremely unlikely that your vote will turn a tie into a win, or a win into a tie, and hence you are almost always better off taking the money.
Poor people are more heavily incentivised to abstain
The (presumably small) payment is going to appear more tempting to a single parent on minimum wage than it is to a millionaire, leading to a skewed effect across the whole population. I also wouldn't rule out candidates trying to promote the payout amongst demographics likely to vote against them in order to benefit from the low turnout.
Low turnouts lead to a reduction in perceived legitimacy of the winner
In a democratic system, the winner wants to be able to point to the election results as evidence that the public supports their policies. Winning a plurality of the 10% of people who actually voted doesn't really show this.
An obvious point, but you're going to need to pay a lot of people. Approximately one third of UK registered voters don't vote, and that number is sure to increase under your scheme. If we pay £10 each then that's over £150 million (and probably rising) per general election, more if we're also paying not to vote in local council elections and the like.
Let me rephrase your proposal. Every eligible voter gets a $10 stipend. In order to vote, you must pay $10 (forfeit the stipend).
This is a poll tax, and the US has a long history of this. To summarize, this approach to provide monetary penalties for voting is generally viewed unfavorably because it disproportionately impacts the poor (who the money would benefit more).
In the US, the 24th amendment forbids this practice.
The right of citizens of the United States to vote in any primary or other election for President or Vice President, for electors for President or Vice President, or for Senator or Representative in Congress, shall not be denied or abridged by the United States or any state by reason of failure to pay any poll tax or other tax.
The government paying (crediting) nonvoters is the equivalent of taxing at the polls, and would have the same results.
An interesting idea which would, however, lead to undermining the democratic system in the end. A democracy justifies its claim to power by saying it is the will of the people (whether that's actually true or not is beyond the scope of this topic). Now imagine that only, say, 20% of the population actually go voting, as the rest simply decides to take the money (since most people realise that their vote, given the total number of eligible voters in the country, is unlikely to have any noticeable impact anyway, so they might as well grab the money) and the ruling party thus amassed a little above half the votes. How could the government claim to be ruling in the name of the people if only a tenth of the population actually supported them? It is also likely that the poor would be more incentivised to take the bribe, meaning further class divides and allegations that the so called democracy is actually just a thinly veiled oligarchy.
Another issue is then that only people passionate about politics would go voting, meaning the voting base would be chiefly composed of activists, who tend to have rather radical views on politics, and who would thus be pandered to by the government instead of the rest of the population (as they wouldn't be part of the voter base and thus would be irrelevant). A democracy would be unlikely to last under such a system, as one thing you likely have noticed about activists already is that they tend to be fanatical in their beliefs and absolutely hate activists of the opposing side - sooner or later, the party in power would start outlawing certain parties of the opposition, citing greater good and similar bullshit.
The proposition seems to be fueled by a desire to end the status quo and bring large scale changes to the system, in which it would certainly succeed and relatively quickly at that (of course how it would be changed is a different question entirely). If that is something you seek, it is probably an idea you might consider supporting. Otherwise, it should be avoided.
Democracy isn't about making the correct decisions, it's about making decisions that are supported by the people. You are mistakenly conflating how much I personally value my own vote (i.e. how much you'd have to pay me to not vote) with how much my vote is worth in the democratic decision-making process. The two are unrelated - in a true democratic system, everyone's vote, by definition, is worth the same amount. It doesn't matter if you're rich, poor, well-informed, or just making a gut decision about which candidate you'd like to have a beer with. Right or wrong, everyone's opinion is valued equally in a democracy. If you want a small cadre of well-informed individuals making the decisions for everyone, then democracy it ain't.
What you're proposing is a prisoner's dilemma variant. Receiving the benefit of keeping your vote depends on other people who share your interests also electing to do the same; assuming a portion will act greedily and take the money, the rational choice is for you to do the same. This is of course assuming the money is an amount sufficient to be significant to you. In practice, it's only significant for the poor, so the effect is to make the rational choice for the poor abstaining from voting.
The reason most democratic systems have countermeasures against notifying who you voted for, is so people can't buy votes. Obviously, the people paying need evidence they actually bought the vote (and you didn't simply steal the money and vote for whoever you wanted anyway).
In the given situation of paying not to vote, how would a voter prove they didn't vote to the 'buyer'? Who voted for what is kept private, and searching millions of vote records to determine who didn't vote would be such a mammoth task as to be rendered not possible. There's nothing stopping a voter from taking the money and voting anyway.
From a different point of view, and let me stress that I Am Not A Lawyer, I'm not at all convinced that (at least in the US) such a contract would be enforceable. Just as you cannot sell yourself into slavery, I suspect very strongly that forfeiting your franchise for money is also not something the courts would uphold.
Presumably the payments would be made by an organization connected to one major party or another, let's call them the Republicrats. Then the opposite party (the Demicans, of course) would encourage their rank and file to sign up, take the money, and then vote anyways. The joys of revenge might well increase the Demican turnout.
So, the Republicrat policy would have to be to make payment after Election Day.