I find it strange that blockstream hold up Visa as the holy grail in terms of transaction throughput and the need for Bitcoin to adopt sidechains so they can reach these levels.

So many lines or worthless FUD and yet still not one fucking clue about what sidechains are about.
So sidechains are not about facilitating a greater number of transactions? Do explain. You sound like you are getting frustrated.
I have not performed a bitcoin transaction in over a year. It is not an appropriate solution for any transaction I've wanted to perform over that time period. I pay a .01 BTC transaction fee when I do because it was about right for the service I was getting.
I very much look forward to being able to do minor transactions in such a way that will not compromise the core Bitcoin solution. As soon as a micro-tipping sidechain comes out, I will peg some BTC to it and transact frequently. In this way, one on-chain transaction will represent thousands of real-life transactions. As a side benefit, one will not be able to analyze my behavior just by reading the Bitcoin blockchain.
I use it weekly to pay php freelancers I hire in India. I also have 3 clients whom I invoice in bitcoin. Works fine for me, as I'm sure it does for others.
Your tipping example means that I cannot tip in Bitcoin anymore, I can only tip in the shitcoin that supports the sidechain its run on.
But anyway, get back to my points above. How do investors handle the coins to be pegged? I thought that only the holder of the private key has any claim on a bitcoin. They will need to hand them over, no?
No.
Think of sidechains as a multi-signature. I can assign one Bitcoin to a sidechain by sending it to a special output which will essentially "lock" the bitcoin on the mainchain and use proof of this "deposit" to issue an equivalent share (token) on the sidechain.
My original Bitcoin is essentially held in escrow until I issue a different transaction from the sidechain to another special output. Once enough work is piled on top of the proof that I sent back the coins to the mainchain (my transaction) then I can use this proof to unlock the special output holding my original bitcoin.