To take your coin flipping example,
But this probability distribution will be useless in predicting the
future as in the long run, I will end up with more heads than tails
since the coin's biased towards heads.
This is missing the key point:
This means that two individuals with different amounts of "evidence"
can come up with two different probabilities
Each time you observe the outcome of a coin flip you have more evidence about the bias of the coin than the "past you" that started from an uninformative prior, so of course the "current you" will have a different probability. For subjective Bayesianism, you can update your prior using Bayes rule to give the new prior for the next observation using Bayes rule.
Now before you observe the first coin flip it is not the case that you are completely ignorant of the bias of the coin. It is easy to make a two-headed coin or a two-tailed coin, but it is very difficult to make a coin that is biased, but still has heads and tails, whilst still being symmetrical enough not to be obviously biased. So a subjectivist Bayesian could construct a prior with a spikes at 0 and 1 for the probability of a head and the rest of the probability distributed fairly close to 0.5. In practice, this prior will give better predictions than a uniform prior, because it has more information about the problem. Of course as the number of flips observed grows large, the results from both initial priors will converge.
Another example where subjective probability is "useful" would be betting on horse races (betting was originally a strong motivation for research on probability). Horse races only occur once, so they don't really have long run frequencies, so we can't apply frequentist probabilities to the outcome of particular horse races. If we thought each horse was equally likely to win a-priori the bookies would make a very large profit. Instead, punters use their expertise to judge the subjective probability that a particular horse will win (based on its physiology, past record, the conditions etc.). The more expertise a punter has, the more likely they will win their bets. The bookies of course are likely to be very expert, and also have the evidence from the punter's bets that they can use, which is why the bookies will make a profit.