No, Not Really.
It is the basic feature of an accumulator design that its accumulator will always be the destination of all ops. An ISA that allows all data instructions to operate with memory as target is already a two operand ISA (usually memory-to-memory), thus not needing an accumulator at all.
6502, Z80 and almost all early micros are clean accumulator machines.
There are of course exceptions with reverse accumulator designs, but they are all notably slower as a final memory write cycle is needed, or special controller designs where memory is integrated into the CPU core like a register file.
[See below for one interesting variation]
Notes
It's important to keep in mind that this is all and only about data instructions, not other types (program flow, housekeeping, etc).
Basic Accumulator-Machine Operation
Accumulator machines, also known as one-operand architectures, are implemented around the idea that an instruction will always only carry one (additional) address and that address being (mostly) the source address. Having the accumulator being always on the receiving side is what makes this the simple implementation it is. The basic cycle is always structured as
- fetch data (from memory *1)
- operate on data (and accumulator)
- hold result in accumulator
It is always only one memory cycle, always a read and always at the start of an operation. That simple principle can cover all operations, including loading the accumulator, as that is just a NOP operation after fetch. All left is the addition of a store accumulator handling to complete the data processing part of a CPU. Add PC handling (jumps & branches) and maybe housekeeping (interrupts, flags, etc.) and we have a quite capable CPU design with minimal effort, ready to add as many useful data operations as wanted (*2).
Other (Data) Register Instructions
Having additional instructions to load, store and manipulate secondary registers - which are not a general accumulator - does not break that rule.
In fact, they also operate with the basic cycles as before - which in theory would allow all ALU instruction also to work on data transferred to those.
Except that comes with a hefty cost. In case of the 6502 there are
- two additional registers (X, Y)
- five ALU operations (OR, AND, EOR, ADC, SBC (*3))
- eight addressing modes
which would total a need for 80 additional opcodes. Numbers for the 8080 would be similar.
Unrelated Cases
Neither does having multiple accumulators change the principle of above very simple processing cycle. Although, in a strict sense, this would make it a limited two-operand-machine.
Note that the 8080 does not fall here, as none of the other registers can act as an accumulator. They are only an additional source address space.
Additional Real World Requirements for Data
While the basic cycle does handle all needs, real world applications do require one more data handling instruction type: RMW instructions. That is a item of data can be read from memory, manipulated and written back in a single uninterruptible instruction. A necessity for parallel processing and interrupt handling. Both mentioned CPUs have them in form of increment/decrement on memory - which usually is the most simple implementation. The 6502 implements in addition several shift/rotate instructions also as RMW(*4).
Couldn't RMW be Used?
Since RMW is already present one could as well use them for all ALU ops. Much like the 6502 is already doing for its memory shift/rotate. Except, that comes as a double cost increase:
A basic ALU op with memory reference costs at least (by using ZP) 3 clocks, while the same as RMW would be 5. Using an accumulator OP followed by store would be 6. So the general savings are rather small if not neglectable (*5).
It also needs, like above, a 40 additional op-codes.
There is Only One Real Use Case
Let's get it straight, storing the result of an operation with the accumulator not in it has only one real use case:
- Holding a constant value to be applied on multiple data items.
That only leaves a very limited real world usages, like masking a string, which seem to be rather rare. Sure, they would be a good speedup in a slim loop, but may not have much impact on over all performance.
Other operations are already stopped by due the 8 bit nature of those CPUs. Any meaning full data item, like for matrix operation, would be most likely made of multi byte items (16 or more bits). Those don't fit an 8 bit accumulator, which means it has to be loaded and reloaded anyway.
Not the Same But Close
Then there is one single real world application which comes somewhat close to the idea and that's Renesas (Mitsubishi) 740 series. Even more interesting as it's a 6500 descendant.
The family features an 'X Modified Operation Mode' or more colloquial 'T-Mode' where many ALU operations don't no longer operate on the accumulator but a (ZP) memory location addressed by X:

It essentially enables any ZP location to act as accumulator. Beside adding many more accumulators, it enables efficient multi byte handling in ZP by using X as pointer. On the down side, all instructions incur a 1..3 cycle penalty:
- 1 for read only, like CMP
- 2 for read/store, like LDA (*6)
- 3 for read/modify/store, like all ALU ops
On the down side the accumulator is no longer fully usable when T is set (*7). It's activated by setting bit 5 of the status register, which adding even more overhead code.

Over all a very nice idea and for sure very handy for embedded (*8), it has a limited usability.
*1 - The 6500 is quite basic on that, while the 8080 adds usage of a register file as second operand, which does reduce code size. Then again, the 6500's zero page addressing is of the same purpose. Neither breaks the structure of them being one-operand machines.
*2 - A kind of hardware that has quite similarities to a BASIC interpreter. It needs only a comparably small core for basic instruction handling (often less than a KiB) which can be augmented with many, many instructions and functions.
*3 - The other 3 ALU ops as well as load/store/compare are already available for X and Y.
*4 - Rockwell's 6500 CMOS design also incorporates similar bit manipulation instructions, further improving usability for embedded applications.
*5 - Of course this ratio gets better with more complex addressing modes.
*6 - Note that neither STA nor PHA/PLA are supported by T-Mode. So while string to ZP can be nicely done by LDA/INX, the other way around is not possible.
*7 - In fact it gets a bit weird as some many accumulator instructions, like shift/rotate or BIT, still work on the build in one.
*8 - 740 based controllers have their I/O Register file within ZP.
inc dorinc (hl)for the Z80. Extremely basic accumulator machines may not have this sort of functionality, sure, but most that I've seen do.