Lets assume I have just installed my client. How can I trust the "balances" (I assume you want to store the UTXO set, but details...) when I can not calculate them myself from history data and compare it to the data in the "pruning block"? If I need to do this, someone will have to keep all history data for me to download and verify. Once I verified the data I can discard it. This however is already implemented and its called - you guessed it - pruning.
It removes the need to trust a 3rd party and replaces it with the need to trust the blockchain:
Currently, if you download only a partial chain you must trust a third party for the balance information of the blocks you did not download.
With this, if you download only a partial chain you must trust blockchain technology for the balance information of the blocks you did not download.
That doesn't answer the question. If a new client joins and they start syncing from the latest pruning block, how does the node know that the balances in a pruning block are correct? A miner could mine an invalid pruning block with incorrect balances but then how would a new client know that that pruning block is invalid? How can it determine this without trusting anyone like how currently full nodes do not trust anyone.
Also, bitcoin doesn't work on balances but rather transaction outputs. The way this would work with less complex changes would be to create new UTXOs with the balances. However, this essentially gives miners the ability to spend other people's bitcoin.
Additionally disk space is becoming a lot cheaper and network bandwidth is also expected to increase. We won't be having to manage a full blockchain with current technology, technology will advance and the blockchain should not outpace technology growth as long as there i is a block size limit.