There are two main areas this change would impact: XP gains, and roleplay.
XP
The bonds system gives, at most, 1 XP to a character per session. In practice, this is probably less, as there are likely to be sessions where a character doesn't change their bond with anyone.
With flags, it seems fairly likely that you can do it most sessions. If you don't have a maximum XP from different flags per session, you might accelerate faster. The website you linked does offer the possibility of removing XP gain from failed rolls if you're giving away more XP.
However, moving gains away from failed rolls may help keep XP gains more level across a group, and it doesn't encourage rolling all the time! That said, I don't feel like XP and levels are hugely important after you're around level 6 or so.
Roleplay
One thing about bonds is that they help you take a moment to explore your character's relationship with other characters. If you think about how your character feels, then your actions and decisions will be within that context, rather than whatever's convenient. There's a degree of immersion into your character's experiences that bonds can help, which flags don't encourage. These are explored somewhat in this question.
As the article mentions, flags are designed for a changing cast, where bonds may not make sense. Instead of colouring your interaction with a particular character, flags instruct others how to interact with you, to give you an opportunity to play out a defining part of your character. Depending on the flexibility of the flag, or how you envision your character, they may be ideal.
On the other hand, flags may also end up being more gimmicky. If players are just pushing your button once a session to give you XP, it becomes throwaway or artificial and may detract from the roleplay. Or it may limit your character, simply acting the same way each session, instead of pushing you to think about how your character's perception is changing.
Summary
I worry that it might be easy to write flags that contain your character's change and growth, rather than explore it. However, bonds can also be hard to write in a way that's actionable. I expect that, as with bonds, given a few tries, one would get better at writing flags that give you the opportunity to express your character in different ways.
That said, I think bonds are really useful as a concept, even if they're not mechanically incentivised with XP. If you're trying out flags, I'd still have people choose and review bonds between sessions.
The short version: it takes time and quiet to trigger
Shouting questions at the Princess in the middle of making (or setting up) some other immediate move doesn't trigger Speak From the Heart, but the reason why is buried in a quirk of the way 3rd-party classes for Dungeon World are commonly written by borrowing wording directly from other Dungeon World classes or other Powered by the Apocalypse games.
Basically, the subtext buried within our common understanding of “comes to you seeking advice” is exactly intended and required by the trigger, and so triggering the move requires having a focused heart-to-heart with the Princess about something that matters. Just moving to stand beside the Princess during a melee and asking whether you should stab Orc One next or slice Orc Two next won't earn a +1 forward for your H&S or XP for the Princess.
A quirk of development history
The move seemed very familiar, so I did some mental digging to figure out what was so familiar about it, and I suddenly realised — it’s almost word-for-word a move from Apocalypse World, the Savvyhead's Oftener Right move.
One of the things often overlooked about Apocalypse World's design is that, integral to its successful move-based design, it has commentary for the MC on each move (including the players'), for how to read and fully understand them all, and execute them well. There are parts of moves that are deliberately up for interpretation, but this commentary exists to ensure that parts of moves aren't interpreted based on mistaken understandings.
The commentary on the Savvyhead's Oftener Right move's trigger is (AW 2e, p. 197, emphasis mine):
“Comes to you for advice” means a whole 2-sided conversation, unhurried and thoughtful, about something relatively significant. Shouting out for advice in the middle of a fight doesn’t count, nor does asking the savvyhead whether to dine upon pigmaggot or screwfish tonight.
And this material from another game matters because the Princess's Speak From the Heart is a functional copy from that game. But being copied from the tried-and-true Savvyhead's move without copying the commentary that ensured it worked correctly has “stranded” it apart from an essential part of the move: the confirmation of what the wording is (too) subtly conveying to the reader of the Princess class. It's open for misunderstanding, and misinterpretation of part of the move that isn't designed to be up for that much interpretation.
So that “unhurried, thoughtful conversation” is what Speak From the Heart's trigger is getting at, and what the move requires for it to not be dysfunctional in exactly the way described in the question. The inherent implication that “Comes to you for advice” carries colloquially, and which has been causing that hesitation, is exactly what the move needs to trigger properly: it has to be in person, unhurried, and thoughtful, just like a real conversation we might decribe as “So-and-so came to me for advice yesterday…”, after the fact.
Yes, this means the move can't be shoehorned into the middle of another conversation, or a fight, or a simple group discussion about whether to go left or right or whether to camp now or later — it’s instead a whole thing/scene/personal moment of its own. Go to the Princess, have a heart-to-heart, and open up about your significant troubles what you need advice with, and you’ll both benefit. And it gives an incentive for planning ahead (because then one might trigger the move by consulting with the Princess), instead of always doing everything by the seat of one's pants in the heat of the moment.
The Princess inspires calm and resolution with many moves, and this move is no different, just more subtle about it than some of the others.
Best Answer
Personally I've never found the starter bonds for classes to be all that compelling for exploring a party's inter-relationships. And for new players they can lead to some poor assumptions about what constitutes "good" ones. To alleviate that:
Add something actionable
Like you've noted, bonds like the Thief's "_____ has my back when things go wrong" tend to just describe states that two characters are in rather than offer them something actionable to draw upon in play. My go-to solution to prevent this is to simply have them add what their character intends to do about it to the end. Bonus points for making it something that can be done in the short term (i.e. next session). For example:
You'll notice that many starter bonds have this element in them anyway, but in my opinion, all of them should. Doing this will also set up good habits when developing bonds of their own—so long as there is an element to them that can be acted upon, character relationships can grow and develop naturally as each one is resolved.