I understand bitcoin transactions can have multiple inputs/outputs but I'm intrigued on how payment providers confirm the receipt of payment in a 3rd party wallet?
A unique address is given to the payer. Payment is not received until the correct amount is sent to that address. Bitcoins sent to that address are assumed to come from the payer and nobody else, since only the payer is provided with the address.
I understand that once you are inside a wallet, you have a different perspective of the transaction and therefore you can provide a clear view to the user about that specific transaction - as you know which addresses belong to that wallet.
But, is there a way to guess, or get closer to an aggregation of the value of the transaction that doesn't belong to a wallet you own? So we could show it clearly to the user: How much was sent, from where to and how much. Ok, there may be multiple outputs, which we have to show, but we can aggregate them and show the sum of the transaction. In the end of the day, once you send a transaction id to a recipient (as proof of payment), they want to see your wallet, their wallet, and the total amount which was paid. Not 15 inputs and another 29 outputs, confirmations, and all that stuff.
All transactions are 100% public. All the information that you are referring to is viewable by everyone.
But I'm trying to put together an explorer where it would be easier to show (to a Bitcoin normal user) a transaction in a simple way (without all those inputs and outputs), but something simpler as From, To, Value, Fee, and that's it.
I understand that, by design, the blockchain doesn't work that way. In any case, I would like to ask the community here if you have any clue or idea on how to achieve this or closer to this

Many block explorers do provide a simple view of transactions (when they are simple), but the wallets themselves are probably sufficient for what you want. The payer's wallet will show info about the transactions sending bitcoins to an address, and the payee's wallet will show info about transactions sent to the payee's addresses.