Saints Row IV is a delight, while I find that on one hand, it pulls too much from its predecessor, it goes in new, crazy directions both on the story and gameplay side. Some of its systems are so amazing and useful that they make others feel obsolete and perhaps, unnecessary. The end result is still fun and refreshing and pulled me to try and get every single thing complete between the story missions.
After a bit of good, old fashioned non-sense, you become the president of the united states. This has close to no implications in-game besides that it helps set up the big plot of the game; You're mostly stuck in a virtual representation of Steelport from Saints Row The Third and you get weird super powers as you go in the game. These super powers range from ice blasts to super speed to super jump and being able to glide, throw things with your mind, and even more. This is what makes this game amazing, being able to do tons of crazy thing inside the engine and everything works more or less flawlessly - although here and there, they let you run alongside walls that end up in ceilings you can't jump through, and it's just weird. But as a general rule, it's really well implemented.
This game is full of legacy systems from SR3, you can have a crew that follows you around, but you'll still beat the hell out of everything single-handedly. You can ride a car; but you'll go faster than all of them and cars can't jump/glide/punch. You can use guns, but why can't you just brawl your way out to victory or use projectile blasts or something? Just remove the guns, make this game more of a brawler/special-powers focused open world game. You can still have upgrade paths for the powers and things like that, but I find it weird to need to collect ammo for my auto-shotgun when I could just be firing energy out of my hands, you also get blast powers after a while and they're really useful, although on long-ish cooldowns. This game could've removed a bit of the 'gangster street life' theme and added more of the 'super hero theme' on the combat side and I think it would've been a bit better.
Steelport changed a bit between the two games, now you have huge towers to climb and alien tron-like things to take over, it's still largely murder missions, races, insurance frauds - even tho they're not crazy since you can throw yourself a billion feet into the air and run at super speeds - mayhem missions where you use tanks and airplanes or other weapons to rack up points - they've added some other missions where you need to jump on platforms, throw things at targets and race around a track, collecting orbs along the way. These missions are fine, but sometimes the controls are too floaty to perform them as easily as you'd want. Maybe the map should've been a bit different... I remember SR3 enough to see that it was quite similar, but you navigate through it so easily that it's easy to overlook.
Another small issue I had with the challenges was the way they integrated in the world. I loved the flashpoints (basically, kill everything in a area) or the tower climbing sections, or the virus injections (kill waves of enemies, sometimes they're aliens, sometimes mascots, sometimes glitchy messes) but the missions having me load out, load in the mission, complete the mission then load back out to the world kinda bothered me, I would have rather be able to start them and seamlessly complete them before going on with my things. I tried to do all the 'instant' missions first and left out the others for later just because of the way they broke my momentum
The story missions are well realized and the game doesn't stop taking jabs here and there at other videogames. Retelling everything that was amazing would be too spoiler-ish and wouldn't help a look at the gameplay systems of the game, but I'll say that you do tons of different things and they're all executed more-or less well. I just wish you had more crazy different things in the random missions, instead of 'kill that guy' and 'run this race'
You unlock stores by playing a little hacking minigame - but if you're getting shot at while you're trying to solve the puzzle, you might die, and there's no way to back out of it, so better think fast - that's basically connecting two sides of a grid with pipes so water flows from one to the other. It's fine. Sometimes the puzzles are plain easy, sometimes they're hard, there aren't a ton of them and unlocking stores net you money over time - also access to the store, but unlike SR3, I never re-customized my character, for the whole game. In Saints Row 3, you were just a dude or a lady and you were walking around doing normal things, sometimes it felt natural to take a break and go change your clothes or hairstyle, get a tattoo or repaint your car, now you just fly everywhere and run crazy fast at oncoming traffic, there's never a good time to go shopping for new glasses.
The upgrades are very similar to SR3, more health, more ammo, nitro on your cars, upgraded homies, more experience, more money... The usual, maybe a bit too similar. You also can upgrade your powers with orbs that you find everywhere on the map. It's fun to collect them and you get a way to see where they are after a while, but I would've loved it if you were told how many there were in a sector of the city, so you could be sure you have them all earlier on. You use these orbs to improve your super powers, your blast has a bigger radius, you run faster, jump higher, can glide, can run on water, etc. All of these upgrades are really fun, even though some can only be unlocked after you complete side-missions or challenges. Maybe hiding them until you unlock them would be a good idea? Can't say it was useful for me to see that one upgrade was locked behind a challenge if I couldn't complete the challenge or side-mission at that moment
Saints Row 4 is great, ridiculous and fun as hell. The ease you have for moving around the city is amazing and the superpowers are all very interesting. I wish they would've scaled down with the 'normal' open-world stuff, but it doesn't remove much from the experience.