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I have seen some players use action surge to cast two spells and your not having seen them but as defense is a mater of play style and spell selection. So basically your punishing people that don't play the way you want/expect. That's a nurf not just on the class but player creativity. Making it an exception goes back to my point that the only purpose of doing this is to prevent/punish multi-classing but we can come back to this in the next part.
If you don't like multi-classing don't allow multi-classing don't allow it then limit there creativity or punish there choose by removing the reason the picked it. As I said, a 1-3 level dip is how multi-classing works best in 5e especially for casters because they lose high level spells and possible spell slots (in the case of Fighters and other non-casters) on top of forcing later class progression. So big loss needs to receive a big reward or multi-classing is always a gimp. If your only allowing multi-classing as a gimp then you should not allow it at all since it will only hurt the players and make play painful.
I get this from your design. You seem to want your players to suffer. If they are good with that, sure, draw out combat by reducing nova. But if that's the case I hope all your encounters are multiple smaller enemies because increasing fight length by reducing player nova into slower damage means single targets that do a lot of damage get more rounds and will do a great deal more damage to groups. This is a good way to TPK your group because you have decimated balance as intended. Again if the players are ok with longer combat and you use roughly a number of enemies equal to the party to design your encounters for a gritty hard fight style that will be fine but if the players didn't agree to this type of campaign they may feel like they showed up to play a fun heroic campaign but instead feel like they are being drug through the mud. By this I am guessing your also playing a low magic campaign and your more combat then story due to the amount of time combat takes. Right?
I disagree. Removing the option to not allow it to be used for spellcasting is not punishing your players, unless you remove it after they have it.
If he (and I) don't want to allow spellcasters to use Action Surge to cast spells, and that rule is known before you choose to multiclass, then it's just a rule. Personally, I don't like the crazy amount of attacks it gives a fighter as written either. So it's very similar to the OP proposal. I did include several actions, but not all of them. I still want the Rogue's ability to be different.
From a game design prospect, I think that the RAW is just fine. It's not
breaking anything. From an in-game-world design, though, it just doesn't fit my world. So it's changed.
Changing it isn't punishing anybody, nor is it limiting their creativity. It just means the options are different than the RAW. If you felt that the options weren't optimal for what you want to do, then just don't multiclass. But that doesn't mean that everybody will feel that way.
Multiclassing exists in my campaign from a character development perspective, not as a "game within the game" to find the best combinations, etc. although you're welcome to do that.
It's specifically for characters (people) that change their path in life. Perhaps they grew up an orphan in the streets and survived by being a thief. Then they meet a mentor who takes them under their wing to train them as a wizard. Some characters in our campaign have three or more classes. It's all based on the story of the character (written by the player over the course of the campaign). It's not because they say, "if I train to be a fighter for a little while then I'll gain the ability to cast two spells in 6 seconds, instead of one."
That's because within the game world itself, "action surge" isn't a thing. It's just a representation of those periods in time where a well-trained fighter gets the upper hand for a moment, and gets in an extra strike. Casting spells in my campaign take longer to cast, and longer for the magical energies to "recharge" to cast another spell. Therefore, finding an opportunity to make an extra strike wouldn't help the spellcaster.
These aren't the only abilities that I've changed, and they aren't changed specifically because of multiclassing. But it will obviously affect what you might be able to do if you do multi class.
You're correct about players agreeing on a type of campaign. That's the case anytime you are sitting down and not running the RAW. I have a pretty hefty book of house rules, really it's the PHB rewritten for my campaign. So you'll know what you're getting into if you decide to play. On the other hand, my "session 0" for a public game is usually a quick adventure with pregenerated characters so you can see how the rules work, rather than just reading them. I find that most people who tell me they won't like a particular rule change, find that it makes sense and works well in the context of the other changes. More importantly, I'm hopeful that the changes allow me to run a game that they find fun and really enjoy, and want to continue playing.
In my campaigns, combats
can take longer, but that's for a lot of reasons, not just lack of the ability to nova. The number one reason is that I think it's absurd that so many combats are finished in less than 30 seconds of in-game time. Also, I would place RAW with their special abilities to be closer to a super-heroic campaign, than a heroic campaign myself. My baseline for a "heroic RPG" is AD&D (1e). Everything since then has been piling on more and stronger abilities. I'm OK with a lot of them, but many need a bit of reigning in to work with the feel of the campaign I've been running for 30ish years. Combats also take longer in my campaign because AC is typically better than RAW and armor provides damage resistance too.
But combat itself is a pretty small part of the campaign overall. In the current adventure, other than the possibility of random encounters, there is only one (expected) combat in the first third of the adventure. That's planned because they are specifically hunting a monster. They will have plenty of opportunity to plan that attack, and they will need it because it's very tough. A good chance of several PCs dying, and I suppose under the right circumstances it could be a TPK.
The second third has no real combat encounters at all (just nuisances), and the final third will have maybe three. Those are tough, and intended to be. They'll need to fight smart to survive.
After that there is a potential encounter at the end of the adventure, but it's pretty much guaranteed death if they are stupid enough to continue, and that should be pretty obvious to all of them. I anticipate each third of the adventure to take at least three sessions, so maybe 4 combats over 12 or more sessions. Plus the potential for a couple of random encounters.
However, some of the other potential directions they PCs can head in have a
lot of potential, even probable, combat. (The initial adventure is an introduction to the campaign). It's all dependent on their decisions.