Yes, but also without the "merge mining". I propose to make sure that the protocol gives convenient access to an atomic-swap market. In my impression, that should be enough to make the different sidechains look like one. As long as the wallet software does the hard work for you, of looking for atomic swaps, carrying them out, and then making the payments within the various sidechains, almost nothing changes for the user.
This is an approach which could also be achieved with a main chain. Apart from that, your proposal looks a lot like "merged mining" if we consider the "shards" to be independent chains.
For the miners, not much will change either. Instead of mining one block in one chain, they will mine 256 blocks in 256 subchains. Of course, they have the option of not mining for one subchain and only to mine for other ones, but why would they do that?
For that to happen, the shards/sub-chains would have to be totally synchronized and the difficulty of all chains would have to be identic at all times. In all other cases, there will be times where it's more profitable to mine on a sub-group of sub-chains or even on one single sub-chain.
I think it's highly unlikely that the synchronicity of the sub-chains will be stable enough to not lead to some fluctuations in sub-chain difficulty. If the difficulty is forced to be equal in all sub-chains by the protocol, the risk emerges that one of the sub-chains still eventually could receive less mining power and then have longer block times. This chain would then become unattractive and the value of the "sub-chain coins" would fall.
Furthermore, if you are an atomic-swap server, you had better watch out for offering irrational prices or discriminate against arbitrary sub-chains, because automated arbitrage bots will not hesitate to take you to the cleaners.
Now imagine an attack on the atomic swap market by a big whale who could profit from it, driving the price of one of the sub-chains down for a longer time and create a loss for arbitrageurs who cannot sell their "cheap coins" for a higher price anymore (and taking also mining power away as the sub-chain coins aren't worth the same, creating a "domino effect").
There
must be a strong peg mechanism, otherwise the peg eventually will be lost. NuBits, an early "stablecoin", for example, relied almost entirely on arbitrage bots, but the peg was lost in 2016 for the first time. It recovered (mainly because the coin developers injected lots of money) but now the coin has
collapsed.
So the path to solve that would be a two-way peg like in the Drivechain proposal. I think it could be achieved, but without a main chain it would be much more complicated as all sub-chains would need the capacity to block coins when they're to be created on other chains.
Even though
Tier Nolan introduced his multi-billion dollar idea of atomic swaps in this very forum in 2013 already, it still took
until 2017 for someone to actually write the first prototype. In my impression, Tier Nolan's atomic-swap idea is much, much bigger than just trustless trading of altcoins. His idea will undoubtedly turn out to be gigantic. In my impression, atomic swaps can almost trivially solve the pernicious problem of sharding and allow for infinite scalability.
There was an implementation in
2014, but not for Bitcoin.

The problem of your proposal is mainly that your rely totally on market forces to achieve price pegs between the sub-chains. That
can work, but it's not guaranteed at all and the system could collapse very fast (like NuBits). Perhaps there is a solution for that, a new form of two-way-peg mechanism, and then you could have found an interesting technology (although I think @aliashraf is right with his criticisms, too). But this problem has to be solved.