RE5 definitely does not have the same issues as 6. The most glaring problem 6 has is that you spend the vast majority of the game fighting dumbass, annoying commandos (albeit mutant commandos) with guns. You spend most of a numbered RE game in firefights. That sucks. Yes, the gameplay is a lot more nimble and versatile than most other TPS game so it's hardly a stock-still cover-based game, but it's still lame, especially when the zombies in RE6 are a lot more fun and satisfying to fight. RE5 does have you fending off against some gun-wielding baddies, but they're relegated entirely to some of the chapters at the end of the game, and not even all of them.
RE6 also has a horribly unfun and inefficient inventory/weapon system that's an embarrassing step back from RE4's and 5's in every conceivable way (no item box, no merchant, weapons are locked on a per-character basis, no way to store/retrieve weapons in any way), and it severely hurts replay value too. The general pacing and encounter design for a lot of RE6 is just terrible too. There's some less good fights in RE5, but nothing even close to the misery of RE6's boss fights against tanks and helicopters, the cricket snipers, god awful jeep racing section, stealth segments, the convoluted as fuck battleship hanger area. The only distinct areas where RE6 unquestionably improves on RE5 is by giving the AI partner infinite health and ammo, and the increase in overall game length is a plus just in terms of raw numbers (even though so much of that extra playtime is a mistake).
The movement speed, the responsiveness of the movement itself, the shooting (firing rate, feel, aiming), counters, dodging, firing from the ground, rolling, quick shot, camera distance, control layout, the block life system, being able to heal on the fly by storing herbs in your case... basically everything pertaining to actual gameplay is better in 6. It goes far beyond the AI. Simple things like your character not having to stop to pick items up or climbing and dropping down being the same button as sprinting are huge. The cross reticle is much better, and it always stays centered on the screen. In 5, you aim and the laser goes around everywhere, it's just awkward. The removal of weapon upgrades is fantastic, and so is the removal of the store. Finding weapons in game is far better. The addition of the skill system is also a nice touch, and adds greater depth as you can set yourself up for certain situations much better.
The inventory system in 6 is very good in that it allows you to quickly access everything (a much greater amount of things than you could carry in 5). It would never work in 5's system, and you can't pause the game like 4. I do agree that 5's setup was the best, and you should have been able to set up quick select buttons and store weapons you don't want to carry all the time. It would also be nice if any character could use any weapon. One complaint I do have regarding 6 is that it just always seemed like you should have one more inventory slot. Carrying all those weapons, but you couldn't carry much for ammo and it made mixing herbs difficult at times as you would have to choose to drop something. As for the series as a whole, I feel that they should have always had the Mercenaries mode set up in a way where you could choose your character and then choose a load out. I've never been a fan of characters being stuck with a specific load out in that mode.
I also feel 6 is much better in terms of enemy design and the atmosphere. It's a dark game for the most part that at least seems to have that horror visual. The Zombies are definitely the most fun part, but the other enemies were very good as well. The mutations and variations were impressive, and it kept you on your toes. 5 was completely lacking in that department, and the enemy designs are some of the worst in the series. I also think the story is worse, though it is more focused. 6 having all the stories running separately and intersecting at points was well done, and it was especially great when the game was new as you would team up with other players at various points online.
As far as having good fights is concerned, I've been playing RE5 since yesterday coming off of 6 the day before, and I'll say this: The worst parts of 6, predominantly Jake's campaign, never go as low as RE5 is for the duration. So we're at opposite ends here. IMO, the gameplay is so inferior that even segments that should be good, the 1-1 executioner (first boss), are more of a chore to play. The slowness of firing and movement, and the clunkiness of the movement hamper the experience tremendously. SIlly things like when he goes to swing his axe you have to walk backwards slowly, quick turn and run, or run past him and turn around, are simply not intuitive. RE5's gameplay simply hasn't aged well at all. This is evident having played 6 just before it. Even normal encounters just aren't that fun because the controls are a huge step back from 6, and just navigating movement on the field around enemies or aiming is just not very good.