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I don't think that's a problem, though I do think there is an error in how the game explicitly defines Straight Flush for the player. If you hover over "Straight Flush" on the Poker Hands screen, it says: "5 cards in a row {consecutive ranks} with all cards sharing the same suit."
Part of the fun of this game is seeing how the poker hand definitions go weird when some of the implicit assumptions that they rely on start to break down. Most basic example is that, in nearly every variant of actual poker (i.e. all that are played with a single standard 52-card deck), there's an implicit assumption that two or more cards of the same rank can't be the same suit. In a real poker game if you have two copies of the Nine of Diamonds, you don't have a pair of nines, you have a misdealt hand.* So Balatro explicitly defining a pair as two cards of the same rank means it starts out as a match to the real poker hand, but begins going weird when you start manipulating the deck.
Defining Straight Flush as something to the effect of "Played hand includes both a Straight and a Flush" would do the same thing. That matches the IRL poker hand definition, before you start manipulating how "Straight" and "Flush" get defined with Balatro's Jokers. And it seems to be how the game is currently defining Straight Flush under the hood. It fits the theme and fun of the game, so I'd say it probably isn't a bug. But I do think the text string on that Poker Hands screen should be changed to more accurately tell players how the game defines a Straight Flush.
*(Unless it's a Yogscast poker stream and nobody at the table notices! 😆)
(edits to correct typos)
Even if Four Fingers explicitly mentioned Straight Flush, based on the text given to define Straight Flush on the Poker Hands screen, OP's hand wouldn't have qualified. Based on that text, the expectation would be that Four Fingers adjusts the Straight Flush definition to "4 or 5 cards in a row {consecutive ranks} with all cards sharing the same suit."
Looking purely at the text currently in the game, that Three of Hearts, in a hand otherwise full of Clubs, means the hand shouldn't count as a Straight Flush, even with Four Fingers. Ac 2c 3h 4c 5c could be a straight, and (with Four Fingers) Ac 2c 4c 5c could be a flush, but the hand isn't "cards in a row with all cards sharing the same suit" so it shouldn't be a Straight Flush.
The game thinks it is, because its internal logic defining a Straight Flush doesn't actually match the text defining a Straight Flush for the player. Its internal logic just checks whether the hand is both a Straight and a Flush; if the hand meets both criteria it gets counted as a Straight Flush.
That internal logic is more interesting, more fun, and better matches the tone of the game, than the text definition. But there's an old saying about the only difference between a bug and a feature being documentation. I'd say this technically is a bug, because results don't match expectation, as set by the documentation. But in this case the fix should just be to update that documentation; i.e. change the text string on the Poker Hands screen, to better reflect how the game actually defines a Straight Flush.
There are plenty of cases with just the standard poker hands, where a given hand might be counted as multiple different hands. In those cases you default to the highest rank.
For example AAAKK could be five cards scored as a Full House, it could be AAA scored as Three Of A Kind (with two un-scored cards), or it could be KK scored as One Pair (with three un-scored cards). Full House is the highest ranking option, so that's how AAAKK would get evaluated. That's a no-brainer in actual poker, but sort of turns into a key mechanic in Balatro, where you may have a lower hand rank upgraded to give more chips than a higher one, thanks to the Planet cards.
Based on how Straight Flush is defined on the Poker Hands screen, and using the logic above, I think it would be fair to expect OP's hand to be evaluated as a four-card Flush, with the Three of Hearts un-scored.
That seems less interesting or fun to me, so I suspect the actual result is the game working-as-intended. It's still a bug though, in a sense, because "working-as-intended" isn't the same as "working-as-expected." But the fix would just be to re-align intention and expectation, by updating that definition on the Poker Hands screen.
A better example is what happens when you play 5 clubs to make a four of a kind. Pretty sure the 4 of a kind wins out there over flush, so it's whichever hand is higher on the initial list of hands that wins out, even if you have leveled flush a few times. Or at least that's how it worked last demo for sure, I don't think I've run into a similar scenario this demo.
Agreed that Full House isn't a terribly notable example since it isn't a Balatro-specific "edge case," and comes up every time you play a Full House. But it's still relevant in that it demonstrates the behavior. And, specific to Balatro, there might be reasons you actually prefer to score Three Of A Kind instead of Full House in a given run; so the fact that Balatro always evaluates the hand as the highest applicable rank is something players build familiarity and expectation around, before they do run into the stranger edge cases.
I didn't feel like that was worth a thread of its own but it did play through my mind a few times, and this feels like as good as any time to bring it up.
In standard poker, no, a flush is harder to make than a straight.
Consider that, for each card in your hand of a given suit, there's one less card of that suit in the deck you're drawing from. Imagine I'm just dealing you five cards, one-at-a-time. If I deal you one Club, the odds of the next card being a Club are 12-in-51, then 11-in-50, then 10-in-49, then 9-in-48. Any non-Club busts your flush.
Say you're going for a straight and I deal you a 5 as the first card. The second card can be any A,2,3,4,6,7,8,9 without busting your straight. That's 32-in-51. The math gets trickier from there, since it will depend on what the second card was (ex: an Ace eliminates A,6,7,8,9 as options, but a 4 only eliminates 4 & 9). But if you work out all the permutations it's still likelier that with five random cards you end up with a straight than with a flush.
Balatro certainly has situations where a flush is likelier, considering you can add/remove/modify cards in the deck, have wild cards, and can re-define what "Flush" and "Straight" even mean. But at the start of the game with a standard deck and no Jokers, the flush should still be harder to draw than a straight.
Before Jokers, card modifiers, and altering the deck come into play, the game is basically just 8-card Draw Poker. The fact that you're picking five cards to play from a hand of eight, and can re-draw up to five cards, makes it easier to get any given type of hand than if you were just dealt five cards and had to make do with that. But the effect applies to Flush and Straight to the same degree, it doesn't boost the odds of a Straight any more than it boosts the odds of a Flush. Extra cards means extra chances to make your hand; but making a Flush is still harder than making a Straight.
The first table on this site gives the odds of making each 5-card hand out of 8 cards: https://www.durangobill.com/Poker_Probabilities_8_Cards.html
Hand ranks are exactly as they would be for any other variant of poker. The ranking doesn't change with more cards.
That it's draw poker ultimately just means more cards. (Re-drawing up to five cards essentially just means you have up to 13 to work with, instead of 8.) Hand ranks won't change; higher ranking hands will still be less likely than lower-ranking hands.