Is the reason why series can't be thought of as simply the sum of partial sums of a sequence because you can't make a group?

So before learning calculus II I had went over the tiniest bit of abstract algebra for other reasons. Currently, I'm using Paul's Online Notes and one thing that Paul is constantly trying to drive home is that while a definite series can be thought of as just a partial sum, an infinite series should not be thought of as an infinitely large partial sum. He gave a variety of reasons why in many of his note sections and to me it seems like the reason why series can't be thought of as an infinitely large partial sum is because the "addition" operation is missing a lot of properties that normally exist.

1: the infinite addition of a sequence is non associative (If you have a series that is convergent but not absolutely convergent then you can rearrange the terms of the series to equal any number you want it to be)

2: There's no guarantee of an identity element in the set that contains the terms of a sequence

3: Addition on the set of a sequence is not guaranteed to be closed

4: There is no guarantee of an inversive element in a sequence under addition

Would the fact that these guarantees don't exist make it impossible to treat an infinite series as an infinitely large partial sum of a sequence because when you create an infinite series it doesn't result in the creation of a group? If that's the case then is the "addition" that is used to generate an infinite series also just straight up not regular addition and is a different operation?

Sorry if these questions are poorly worded, it's 7 in the morning and I'm in a physics lecture so I'm mentally exhausted lol.

Thanks in advance