Hi there and welcome!
The answer is that YES it will definitely be an improvement over your cast iron and oven approach.
I use propane over charcoal grills because I simply do not have time to start a fire, get the fire right, let the fire burn out, and then clean ash from the grill before the next use. A charcoal grill does impart better flavor and I would never argue otherwise, BUT I will argue all day that it doesn't impart such better flavor that it is worth the extra time (to me). Gas grills produce mighty fine food! I like to think of charcoal vs propane the same way I look at using salt. Kosher tastes better than table salt but it's not worth my time to go to the store and buy kosher if I have table salt in front of me to use
On a good entry level 2 burner propane grill I can have amazing steaks for 2-6 people cooked from start to finish in 15 minutes. I start the grill and go inside and prep the steaks (pull from fridge and season). When the steaks are done being prepped, a GOOD entry level grill on high is already searing hot (over 550F) and I throw the steaks on.
I do 3-4 min on each side for medium rare to medium and pull. I turn off the grill
and close the propane tank off.
Take everything inside and eat while the grill cools off.
Clean up dirty dishes and save left overs and the grill is cool by the end of th emeal. I now walk outside and cover the grill.
The whole ordeal from beginning to end can be as fast as 45 minutes!
Since your apartment complex allows a propane grill I would encourage you to get one, 110%.
Now the key is to get a propane grill that gives you 80-100 BTU's per square inch of cooking surface! Most gas grills, expensive or cheap, often fail to meet this simple requirement
I bought my girlfriend an entry level propane grill that does 100 BTU's per square inch for $75 brand new back in 2016 and MAN I love that little grill for what it is!!!
Unfortunately you can't get it for $75 anymore but I found it here for a decent price when compared to the other grills that are more expensive yet less performant AND I can vouch that this grill has everything you need to make a great steak as it gets super hot and it was pretty simple to assemble. It lacks elegance, bells, and whistles but overachieves in performance and functionality and that is what really matters here:
Another less expensive entry level option that also meets the 100 BTU per square inch requirement is this Huntington grill:
I believe Huntington usually comes through on the BTU's because I bought my current home grill on craigslist because it met the BTU requirements and it just so happened to be a Huntington... but a much higher end grill than the entry level one in the link.
If I were in your shoes or recommending a starter propane grill I would not hesitate at all with the Thermos (1st link).
Seeing as how I have a higher end Huntington at home I would also give the entry level Huntington (2nd link) good consideration.
With any entry level grill I would also invest in a long grill lighter as you never know if the push button ignition will work, or for how long it will work. My girlfriends still works to this day. Here is an extra long grill lighter that should last a while:
I hope this info helps and I hope you get to grilling really good steaks and other food soon!
