Unfortunately most games aren't complete as you purchase them. Whether it's patches, DLC, whatever. As a collector, you have to judge whether to skip out on games for that very reason or purchase them wondering if there's going to be a more complete release later on. However, very few games are online only or even required.
Payday 2 got recently released on the Switch and if you were to buy Payday 2 The Big Score on the PS4 and Xbox One, or even Payday 2 on the Switch, it's an online co op game(or offline co op for the Switch) with 55 different missions. If you decide to play solo though, there's nothing stopping you from completing any mission in the game even on harder difficulties. Your ally AI is good enough, about the only thing they don't do is assist you in carrying bags or doing any of the objectives. The reason I mention this game though is that I do feel that even though this game is a multiplayer experience, once the servers are done, the single player experience will still be able to carry this game along, and as I said before, the Switch has offline co op provided you have four Switch consoles.
Now I grew up starting with the Atari 2600/NES in my household, but I honestly think that gaming is better than it's ever been. A lot of people get it in their head that classic gaming is better because there were no patches, games were complete, and we're well aware of all the quality games back then, but do you know how many games were absolute trash? Additionally, what do games like the early Mega Man games, Super Castlevania 4, and many other games, including popular ones have in common? Lots of screen glitches, slowdown, etc. Secret of Mana not only had framerate issues but it had many gameplay issues, and argue as you might that this stuff was intentional, if not for the great music and graphics, it wouldn't have been worth touching as the gameplay was a very mediocre experience. Metal Gear Solid 3 on the PS2 never rose above 20fps. There are lots of issues for varying different games across consoles.
On the plus side, even at launch, many games today are atleast solid 30fps on consoles. Secret of Mana which had slowdown is 60fps for the remake on the PS4(Vita runs like crap though I've heard.) It's rare that games are constantly crashing, but one that comes to mind is LBP3 which crashed six times before I made it out of the first world, and that's after all of the patches. Now days it's honestly most publishers of indie games that could and should run fine on the console that run worse than the AAA titles. THQNordic is a pretty notable one to look towards with games that generally run like ass, from both Darksiders 1 and 2, to nearly every single indie release they've published that I've played, that being said they also have a decent amount of quality releases.
Now there's one genre that I still adore, but is pretty terrible now days, and that's JRPGs. There are also some gameplay styles that excelled in the past that are now missing here as well, but overall, there's a far greater variety in the gaming space today. Now honestly, we can think indie developers alongside the larger publishers. The larger publishers are always going to go for the casual audience, so the games will unfortunately be catered to them. However, many of the smaller publishers and many indie developers have created a variety of experiences that push the gaming space further than it's ever been. There's still your large amount of trash titles, but that's usually digital only(even though there's also digital only games that are excellent.) Honestly though, most games that you can find at retail are guaranteed to be atleast better than a large majority of the past gen releases.
I purchase anything that interests me and I sell it off if it doesn't hold a certain amount of quality. In my collection, other than the rare release(not actually rare, just it's uncommon that I keep a bad game) that I don't feel I can get rid of, I like to only keep what I consider good games in my collection, so I'll sell off anything that I feel isn't. I also don't buy any online game. I purchased Warhammer Vermintide for example because I didn't realize it was online only, found out sometime later and it's in the smallest text on the back of the box, and despite the fact that I still haven't played it, I sold it. I've only purchased one online game, Destiny, and after I was bored with it in less than a week, was pretty much a coaster, so if it's online only I won't even play them. With all of that being said, I still own over 150 PS4 games. I also buy retail only.
Games like..
Hotline Miami, a stealth/action based murder simulator where you go for a high score.
Furi. A fast paced melee action game combined with the likes of a twin stick shooter.
Has-Been Heroes. A true hidden gen of this generation. Looks stupid and is simple to play, but the amount of underlying depth and strategy has it being my most played on the PS4 to date.
Overcooked. A time management cooking game. You have to work efficiently and effectively to get orders out quickly and correct without setting the kitchen on fire.
N++. A very minimalist platformer with thousands of stages. Unfortunately there's a patch to double the content to 4000 stages. Sad we didn't get the complete version but ehhh, it's still 2000 stages and you can create your own.
Divinity Original Sin. One of the best WRPGs I've played in a long time, better than a lot of the classics even.
Bloodborne. Nightmare Creatures is one of my favorite action games on the PS1, and while there is none(yet, did you guys hear someone is rebooting the franchise) on the PS4 there's still the amazing Bloodborne and Dark Souls series.
This is the Police. A visual novel with light strategy elements. You're a 60 year old police chief nearing retirement. It's a simple game, but there's so many underlying statistics, some slight RNG with your officers, making the mayor's office happy, making the mafia happy, and your goal is to obtain $500,000 by the end of 180 days when you're forced to retire without being put in prison or murdered.
Dragon Quest Heroes. The first game, not the second, is a RPG/Dynasty Warriors/Tower Defense mix. Now I hate Dynasty Warriors, the series is pretty garbage, but the way it's done in Dragon Quest Heroes and the tower defense style of it has you warping back and forth between different areas of the map, making sure that each area isn't overrun, it makes a much better experience than any of the other games. The second game is a bit worse than this one, but atleast has some difficulty to it making it again a bit better than the rest. The only one I like even a little bit after these games, is the first Fist of the North Star Ken's Rage.
Yakuza 0, Kiwami, and undoubtedly 6. Yakuza is an "open world franchise" unlike the open world games that you generally think of. Unlike Ubisoft's open world games, the quests in Yakuza all have some sort of storyline to each of them. The game has a ridiculous amount of activities that you can take part in. Then the single player storyline is always some of the best writing and storyline in gaming. You want to take a break with the main quest and do some side quests, you can, you want to just sit back and play Space Harrier, Outrun, or Super Hang On to try and top(and admittedly fail) the leaderboards, you can, you want to play a game of Shogi, Mahjong, and a lot of other western and Japanese gambling games where the AI are cheating a**holes, you can do that too. And that's just scratching the surface. The game is a beat em up with a great storyline and tons to do.
Just like I only scratched the surface with all Yakuza has to offer, this list only scratches the surface of all the PS4 has to offer. These games really just show a minimal amount of variety that the console has.