The least significant bits are a waste of time putting in. If you have only 1% resistors then you can only get usefully 6 bits of resolution. The rest is noise. So yes it will function but not very well. The problem is one of monotonicity
From Digital to analog converter - wikipedia
Monotonicity
The ability of a DAC's analog output to move only in the direction that the digital input moves (i.e., if the input increases, the output doesn't dip before asserting the correct output.) This characteristic is very important for DACs used as a low-frequency signal source or as a digitally programmable trim element.
So yes it will function, to some extent.
No I am not saying that. I am saying that a MIDI message and a sound are two different things. The fact that they both can use a serial interface to transport them does not make them the same thing. Which is the impression I got from your post.
So it would be possible to generate a waveform from the Arduino using a look up table, but these sounds would be static. That is they would have a fixed, none changing waveform. Note on / off are easy enough but other things like after touch are subtle effects, not always implemented in professional sound generators.
Then there are things like envelopes, where the volume of the note produced changes over time to give a bit of variety in the note. Finally there are the full blown synthesisers of many types you can implement that have their own characteristic range of sound. On top of all this are effects like echo, reverberation and tremolo. This is a big field.
I have used a more powerful processor for this sort of thing, like an Raspberry Pi Pico 2040, and a Teensey 4.1, and there are many examples of implementing synthesisers on line.
I have been exploring the use of the new Arduino Uno R4 Minima and WiFi to do some simple waveform generation lately.
Like turning a Game of Life into a series of tones, see this video.
Your circuit after the resistor ladder looks suspect. What are these 922 amplifiers? Do they have a real part number? First off you have no power supply decoupling on the chips, and second I doubt you can get the full voltage swing out of them. That is there will be a maximum and minimum voltage they will output, no matter what the input is.
Also the volume control pot, apart from having the wrong schematic symbol, will affect the DC level of the signal, causing distortion of the signal because the whole thing is DC coupled. So that bit of the circuit needs work on it.
As it seems you are just getting into MIDI then I suggest you use an Arduino Micro or Leonardo and just handle the MIDI messages with that an put them out to a MIDI sound synthesisers, like the Adafruit Music Maker FeatherWing.