I really wonder whether the new blockchain works as intended.
I do not get the part that each transaction requires the sender to verify two other transactions on the Tangle.
Does this means that a sender is always dependent on future senders to get his payment confirmed (checked and verified by those future senders)?
If so, in theory the system could encounter a problem if 10,000 senders make a payment and thereafter, only 1 person makes a payment each day. It would take a long time before that sender has verified all 10,000 payments before that. And of no one makes a payment for a long time, all the latest payments (still unverified) would remain unconfirmed.
Am I missing something?
They have a mechanism so the DAG is single ended, which means if 10,000 senders make a payment they will verify each other and there is only one at the end. This is a high volume transaction tech used on IoT, which means there is a huge amount of transactions per second. Your payment will usually get verified in milliseconds while on blockchain it takes 3-5 seconds. If a machine plugged into the network is appending his sensor data to the network there is a continuous stream of transactions so the scenario where there is a very long time world wide without a transaction if very much unexpected.
I think you got something wrong here. There can be multiple DAG endings (called tips) in IOTA. The system is designed in a way so that participants are motivated to validate more recent transactions and especially those that have not been confirmed before (=tips), though, else the system would not work. You can view a live visualization of the IOTA tangle at https://tangle.blox.pm/.