I wanted to come in with a little bit of math.
Here's how much damage flurry of blows can do at the different levels with your action+bonus action and 1 ki; keep in mind, misses are 0 damage (this monk starts with 16 dex and increases at 4th and 8th):
3rd: 16.5
4th: 19.5
5th: 30
8th: 34
11th: 38
17th: 42
Burning hands deals 10.5 damage for 2 ki, and you still have your bonus action for other uses (not a big deal, though). Burning hands does damage on a miss. Maybe at 3rd level, against 2 or more targets, it's a good deal, but after that it's really bad. A high level monk does equivalent damage per attack at a point, and can move between them.
The monk's ki points are incredibly close to the amount of spell points a 1/2 caster would have, divided by 3 since they get short rest recovery and days are balanced around 2 short rests. The Elemental Monk's max ki value is also equivalent to spell point costs for every level, except it doesn't have the 3rd level increase. So I don't disagree with the costs for the Elemental Monk's effects, just the action.
1 ki and a bonus action gives 2 attacks for 11 damage at 3rd level. Burning hands deals 10.5 damage (save half) for 2 ki. Against 2 targets, that's equal. Against more, it's gravy. That would be fine. Again, this is exactly the 6th level ability for the Sun Soul Monk.
I suppose if you gave that to the elemental monk, then the Sun Soul Monk is losing some uniqueness and may need some extra options. But when you think of the Monk as a half-caster, then the Sun Soul Monk only giving 1 cantrip and then like 7 spells known, it is nothing compared to Paladin subclasses that get extra spells known AND other abilities.
That's why the Way of the Four Elements monk feels week. Getting bonus action spells would make it feel good.