Imagine, a datacenter network running full nodes & no user running a full node.
Would that be unsafe? If no, why do bitcoin users run a full node? If yes, why don't bitcoin users run a full node?
Would that be unsafe?
In practice, no. There are some theoretical attack scenarios that are extremely unlikely (if not impossible) to happen in practice, and I am not aware of any successful attacks on users running light wallets such as Electrum or Ledger, because it would take more than 51% SHA-256D mining power to fabricate a proof that would convince a light wallet of some alternate reality - but the alternate reality would create a hard-fork with no market value so the attacker would lose a huge amount of money just to convince his target.
These wallets may be "light" as in not running a node but they do not have to trust the server (the server runs a node that feeds the required data to wallets), because the server produces a cryptographic proof of your TX(es) inclusion into a blockchain with the highest work.
From the Bitcoin whitepaper (the below is copied from markdown version found here):
It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he's convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it's timestamped in. He can't check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it.
As such, the verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. While network nodes can verify transactions for themselves, the simplified method can be fooled by an attacker's fabricated transactions for as long as the attacker can continue to overpower the network. One strategy to protect against this would be to accept alerts from network nodes when they detect an invalid block, prompting the user's software to download the full block and alerted transactions to confirm the inconsistency. Businesses that receive frequent payments will probably still want to run their own nodes for more independent security and quicker verification.
Imagine, a datacenter network running full nodes & no user running a full node. Would that be unsafe? If no, why do bitcoin users run a full node? If yes, why don't bitcoin users run a full node?
Yes, it will be unsafe. Bitcoin users do run full nodes but not everyone. Its not possible to convince or force anyone. Benefits of running a bitcoin node are shared regularly and its easier to run a node now compared to a few years back.