The premise of your question is flawed. Much faster RAM than 2.6MHz (which I believe you have picked as an approximation of a 375ns cycle time, which is a critical number as will be explained below) was available in 1981. It just wasn't used because there was no point.
The 4164 series of DRAM chips were launched in the late 70s with speed grades available down to 120ns.
Home computers of the era may not have used the capability. DRAM chips store data in a 2D grid in order to minimize address decoding logic. Basically, a row and a column are selected, and the bit stored at the intersection is active. To reduce pin count, the row and column address are provided separately, controlled by 2 lines, RAS (Row Address Select) and CAS (Column Address Select). To specify an address, you connect RAS to ground while the first half your address is on the pins, then CAS for the next half. Obviously this takes longer than just providibg the address all at once, so if you are accessing two columns in tge same row you dont need to change the row address again. This is called page mode.
Using page mode would have required a DRAM controller capable of recognizing when the next access was going to be within the same row as the last, and would thus require latches, comparators and a fast clock signal to sync to, which was beyond the level of sophistication of the typical home computer, which would be more likely to have a capacitor and a Schmitt trigger to sequence the RAS and CAS lines.
But even without using a proper DRAM controller, the cycle time for a 4164-120 is 220ns. So of course the home computers of the era didn't bother with using page mode: they were mostly using either a 6502/6510 (max clock speed 2MHz) or Z80A (4MHz, but doesn't expect response to its memory requests until 1.5 cycles after it issues them), so you could easily get away with using memory that only responded after 375ns or longer, which is 1.5 cycles of the Z80A bus, and therefore the fastest memory access time that would make any difference in a 1981 era home computer.
And that is why you have the erroneous belief that you couldn't get RAM faster than 2.6MHz -- you could, but it wouldn't have helped because that's the fastest speed your processor would have used it at.
Edited to add - I've just discovered a wonderful resource that can help answer your original question. In the pre-Internet era, it was of course tricky to get hold of datasheets for random components in order to evaluate whether they might be useful, and so on, a problem which led to the production of books that contained useful selections of datasheets and application notes for common (and sometimes not-so-common) components, along with indices to help you find components that might perform the function you're looking for. Sometimes this was just one manufacturers data, but there were also publications that covered multiple manufacturers, including one called "IC Master". It happens that several editions of this publication have been scanned and are available on archive.org.
Unfortunately, I can't find the 1981 edition. But they have the 1979 edition and the 1983 edition, which should give you some idea of what was around in '81.
The 1979 edition shows that, as you might expect, larger ROMs were slower than smaller ones, so it depends how big your BASIC interpreter was. This page contains a summary of the available ROMs in that year. As you can see, for any size larger than about 2KB you will be using one of the MOSFET-based technologies (mostly NMOS, but some CMOS chips were available), which was somewhat slower than the larger and more energy-hungry TTL implementations. So for a 2KB ROM, you could get access times down to 100ns/10MHz (i.e. better than most reasonably-priced RAM chips of the era -- scroll a few pages back and you'll see than a 4Kbit DRAM could be as fast as 40ns if you were willing to pay for it, but most were in the 150-400ns range), but for a larger 16KB (as used in the ZX Spectrum, for example) you'd be looking at 450ns/2.2MHz.
By 1983, the situation had improved quite a bit. 4KB fast TTL ROMs were available, and the MOS-based parts had begun to catch up with TTL in speed, so you could by then get a 16KB ROM as fast as 200ns (5MHz), while DRAM speed was practically stationary (100ns was the fastest speed available for a 64Kbit DRAM in both years, although I imagine it would have been substantially cheaper in 1983...).