Blockchain as a service - is this really just a replacement trusted third party? One of the key tenets of Blockchain technology is that it is a trustless, distributed ledger which removes the need for a trusted (but hackable) third party to oversee the veracity of transactions (e.g. a bank). The original concept was implemented in Bitcoin and depended upon "miners" to verify the authenticity and ordering of transactions. These miners are owned by participants in the blockchain and no one is able to control all (or even a significant number of) the miners in the network, thus rendering the system secure.

We now have the concept of "blockchain as a service" from providers like Microsoft and others, where apparently there is no need for miners: so does this mean that the technology provider just became a replacement trusted third party and if so what prevents them from being a weak (hackable) link which would undermine the whole purpose?
First lets distinguish between a blockchain and Blockchain as a Service (BaaS), because they're two different things.

Bitcoin, and other crypto-currencies, use a distributed ledger in a peer-to-peer network - a blockchain. This is the original use of a blockchain. As you state correctly, the fundamental concept is that it is decentralized, permissionless and trustless. There is no central management, anybody can use it and it does not require trusting in a third party. For Bitcoin, and other crypto-currencies, the peer-to-peer (P2P) network propagates transactions, verifies the transactions as valid, and then adds these transactions to the blockchain.
Bitcoin is the first blockchain application to create a cryptocurrency. Ethereum is a cryptocurrency which is also a platform for running distributed applications (DApps).

Software corporations like Microsoft and IBM are offering Blockchain as a Service. This is something completely different than a cryptocurrency. It's closer to what Amazon provides with it's AWS cloud platform. If you're not familiar with cloud computing platforms get up to speed on that.

Looking at Microsoft's blockchain solutions and IBM's blockchain solutions you can say that BaaS provides tools, infrastructure and/or a simulated network environment for companies to develop their own blockchain applications. These blockchains may be on an established decentralized public platform like Ethereum, or it might be a private permissioned blockchain.

A private blockchain is not anything like permissionless Bitcoin or Ethereum cryptocurrencies. Comparing a private blockchain with Bitcoin is analogous to what the public internet is to a private LAN. It may use concepts like proof-of-work mining or other features of Bitcoin, but it's entirely different thing essentially. I'm not sure exactly how companies are finding this more valuable than simply using a database. Maybe there's good reason or maybe it's largely hype and wanting to get on the blockchain bandwagon.

A private blockchain would require a P2P network, but there would be a restriction on what nodes could access and interact with the network. Also the software would be closed source and not freely available. In contrast crypto-currencies like Bitcoin and Ethereum are open-source and anybody is free to participate in the P2P network.

Going back to the specific questions you had:
Does this mean that the technology provider just became a replacement trusted third party and if so what prevents them from being a weak (hackable) link which would undermine the whole purpose?

As explained already BaaS provides tools and infrastructure for other companies to create blockchain applications. There is no "Microsoft Blockchain". It's up to individual companies to create either public or private blockchain applications however they like. In general and ideally blockchain applications are peer-to-peer and decentralized.

If it is an Ethereum DApp then ultimately it would run on the public internet on Ethereum platform in distributed way without any centralized reliance on third parties. If it is a private blockchain, then it will be a hybrid of decentralized and centralized architecture and not anything like Bitcoin or Ethereum.