In your proposal, transactions go to a chain based the addresses involved. We can reasonably assume that different people's wallet will tend to be distributed uniformly over several sidechains to hold their transactions (if they're not, there is no scaling benefit anyway...). That means that for an average transaction, you will need a cross-chain transfer in order to get the money to the recipient (as their wallet will usually be associated to a chain that is different from your own). Either you use an atomic swap (which actually means you end up briefly with coins in the destination chain, and require multiple transactions and a medium delay), or you use the 2way peg transfer mechanism (which is very slow, and reduces the security the recipient has to SPV).

Whatever you do, the result will be that most transactions are:

* Slower (a bit, or a lot, depending on what mechanism you use).

* More complex, with more failure modes.

* Require more and larger transactions (causing a total net extra load on all verifiers together).

And either:

* Less secure (because you rely on a third party to do an atomic swap with, or because of the 2 way peg transfer mechanism which has SPV security)

* Doesn't offer any scaling benefit (because the recipient needs to fully validate both his own and the receiver chain).

In short, you have not added any scaling at all, or reduced the security of the system significantly, as well as made it significantly less convenient to use.

So no, sidechains are not a direct means for solving any of the scaling problems Bitcoin has. What they offer is a mechanism for easier experimentation, so that new technology can be built and tested without needing to introduce a new currency first (with the related speculative and network effect problems). That experimentation could eventually lead us to discover mechanisms for better scaling, or for more scalability/security tradeoffs (see for example the Witness Segregation that Elements Alpha has).

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Pieter