I wouldn't say that cycling is "bad" for your health. There's solid research that cycling can fix a bum knee, as an example:
It is suggested that cycling might be a useful exercise in the
rehabilitation of patients with injuries to the anterior cruciate
ligament, medial collateral ligament of the knee or achilles tendon.
And just to ground everyone else, the hip muscles in general form a network that is often treated as a group, and iliopsoas is a compound muscle created by the psoas major and the iliacus. The whole web of muscles in there often serve to provide different leverage points. At a certain angle one muscle is the prime mover and then past that angle a different muscle has better leverage, especially in adduction vs abduction: stuff like that. And for a terrific video on the architecture and function of the iliopsoas watch this video. It shows the interaction of anterior pelvic tilt really well.
A humbling thing to realize is that the human body isn't fully understood, and the psoas major is in that camp:
The function of the psoas major is another area of controversy and
uncertainty in the literature. It is well established that the psoas
functions as a primary flexor of the hip joint [...] but it is the
other actions that are not well understood. There are several
hypotheses that have been put forward that are worthy of
I don't know if there's any research linking tight hip flexors to cycling, and it's going to be dose-dependent if that research is done. But anecdotally, tight hip flexors and lots of cycling seem to present together.
I don't think it's worth tar-and-feathering cycling though, because sitting at a desk is a great way to pick up anterior pelvic tilt as well but of course you get none of the benefits that come from cycling.
I would look at cycling through a rather simple lens: it's a great activity and a great exercise, but it's not a whole body exercise and it has body position that's not anatomically "normal". It limits your range of motion and loads a lot more of you in some parts (legs) than others (arms). 22% of cyclists in a study reported overuse injuries in the lower back.
Strength training (squats, deadlift, press) have been a staple in professional cycling for a long time. I remember reading Greg LeMond's book in the mid 90's with pictures of him in a squat rack. I provided an answer to a guy asking about strength training for runners, and I would recommend the same for a cyclist in lieu of a more structured program.
Whether or not cycling is good for lumbar hyperlordosis (and it probably isn't), the way to fix it is the same tried and true rx: strengthen the whole chest-through-knee muscle chain with full range of motion strength training.