I consider the multiplayer updates as free updates because they are free as in, you can play them without paying for them. If you want to pay for anything related to them as in the cash shop then that is optional and by no means a requirement nor are you pressured to, in my opinion.
If you are referring to the additional crates you can buy then those are purchasable entirely through credits gained through sessions meaning that you only pay for them with real money if you choose to. What content there is in the multiplayer updates are avaliable to you for free.
You misunderstand. The Multiplayer updates aren't actually free at all. The only reason you have that "belief" is because the cash shop allows the "free" updates to be possible. Money comes from somewhere, and EA never gives anything away for free. Again, you can look to all the DLC ME2 and ME3 has, where you have to pay money to buy just one weapon or a skin pack. There's nothing "free" about any of the multiplayer as some player, maybe not you, is making it possible.
Alleged malinvestment in multiplayer isn't responsible for ME3's problems. People just don't like it so it's a convenient scapegoat.
A horde mode isn't going to make your writers come up with dumb stuff like finding a magical mystery weapon under the couch or have a terrible character be your "rival." That is just BioWare's inability to hire or retain good writers. The same thing was responsible for the trashy writing in ME1 and ME2.
People complained that their choices didn't matter but that is just the same design philosophy I've seen in every BioWare game I've played. Choices have very rarely been anything more than superficial.
You do not undestand how game development works. Studios plot out a roadmap. They have a list of priorities they want to see get into the game. Depending on time and what they are able to do, this list of priorities will fluctuate and things will either make it in or be cut for something else. MP was a high enough priority that part of ME3's funds were dedicated to the feature and another BioWare studio was tasked with getting it done. It may not have impacted the overall writing directly, but it definitely impacted the development of the game as a whole.
If you believe the writing in all three ME games are terrible, why are you even here? True reactivity and choice is virtually impossible. To account for every action and variable leads to an over-burdensome and unruly development design. BioWare creates reactivity and choice where it can as well as using illusion. It's not perfect by any means, but at least they try unlike most AAA developers. Again, every aspect of development affects the game, including post-release content such as DLC or expansions.