Even with LN, we can't talk of "infinite scaling". We need to open a LN channel for every user with an on-chain transaction, and there must also be space on the blockchain to close it. Doing a little bit of kindergarden math: If we'd 2 MB blocks (~the effective estimated Segwit block size) and a value of 180 byte per transaction, then ~11000 users could open a payment channel per block, or 1,5 million per day. That's a lot, but it's not enough for the whole world's population, and it doesn't take into account channel closing transactions and other on-chain transactions.
That's not taking something very important about the transaction capacity of Lightning Network into account; once the channel is open, you can use the BTC in the channel infinitely (well, unless you run out of BTC of course). There is no need to close the channel, unless you need it for an on-chain transaction, or if someone you exchange BTC on Lightning with tries to start replaying old transactions.
So your "kindergarten math" is not very insightful, because it paints the absolute worst case scenario for Lightning's capacity.
Sidechains and extension blocks are a way to get closer to "infinity". I prefer a three-layer system: 1) Main chain 2) Sidechains or "extension" blockchains 3) LN.
Extension blocks simply produce the same outcome as Bitcoin Unlimited with a different method. The miners control which extension block they put a regular user's transaction into, and so they can create new chains at will, and force users to download the extension chains they create against the user's will.
And so extension blocks simply give miners all the power to turn the mining network into servers that control the client software that the users use, Bitcoin would cease to be decentralised or peer to peer. Miners could create extension chains so huge that it would be no longer practical for regular users to download all the extension chains that their transactions are placed in, or to donwload the chains that contain BTC they receive from others. It's basically an even worse idea than Bitcoin Unlimited, as the miners don't even need Bitcoin nodes voting for the blocksize, they can change the blocksize to anything they like at whim.