I've actually been looking for games kind of like this.
This seemed impossible (I kept dying at the boss attack where the entire screen fills up with the attack) until I realized that not pressing the button would cause the ship's attack speed to increase, at which point the game became a lot easier and better designed. I like the idea of trying to have minimalistic input, both as a game and as a player - doing nothing in order to attack reminds me a lot of Archero.
The biggest issue with this game is that the boss' attacks aren't telegraphed well enough. Sure, you get a warning, but the rotating beams begin rotating at full speed, rather than accelerating (I have no idea why this is, since the beams slowly decelerate.) Normally this wouldn't be an issue, except that the player has no idea whether or not the rods are going to move clockwise or counter-clockwise, which means they sometimes have basically no time to change directions.
Similarly, the big red attacks that are transparent don't give you a good idea of how much time you have until they activate. This is usually not a big deal since you simply get out of the damage zone as quickly as possible, but it can be a bit annoying. More importantly, some of the largest area attacks still seem to give you the same latency before the attack as attacks with a smaller area of effect. In some cases it feels like I simply didn't have time to get out of the way.
If the design of the game is to force the player to play as aggressively as possible so as to finish off the boss before it becomes unbeatable, I think this game is fairly well-balanced. However, I still think that the late-game attacks should be easier to dodge.
As far as suggestions, it would be cool if there were obstacles or pickups that could appear along the player's path. For instance, you could have some kind of obstacle that stays in one place and damages the player, but can be disarmed by approaching from one side. Alternatively, you could have powerups that may appear on the circle. Either way, incentivizing the player to do things that aren't only related to avoiding attacks could give the game some more risk and reward mechanics, as well as allowing the player to express some autonomy beyond just dodging.
Regarding the alternate control scheme, I'm not a fan of the boosting ability. The attacks require you to boost so frequently that you run out of fuel and are forced to take damage. Not being able to change directions takes way too much control away from the player.
If you're just playing around with different kinds of ideas, I have a game called Chernobyl which has a 1-button control scheme that you might find interesting - the player has the ability to navigate the entire 2D space, rather than simply changing direction as is the case here.
Some good potential here, but I think you were right to get a second opinion before going ahead with a full game.