Speed/pitch accuracy has always been of concern to me (even though I'm not in possession of absolute pitch.)
You can always trust a 1KHz tone generated from a computer to be exactly the right frequency, because the sample clock is derived from a crystal oscillator.
I've never seen an (old) console that gave an accurate 1K tone. (My old Studiomaster's was something like 1300Hz!)
That tone is only intended for setting levels.
MRL states that the frequencies on their tapes are accurate to within 0.5% - the frequency on the tape should be between 995 and 1005Hz. The speed and flutter (3150Hz) test tape is even more accurate.
What kind of machine do you have?
If it's very old it will have an AC synchronous motor for the capstan. The speed will be governed by the power line frequency (50/60Hz), which is generally quite accurate. In this case there isn't anything you can do about the tape speed besides build a variable speed oscillator (connect a sine wave oscillator to an amp large enough to generate a 120Vrms waveform and run the motor off of this - I haven't ever had the occasion to try it!)
More likely your machine has a DC servo capstan motor. The motor has a frequency generator built into it and this frequency is compared against an internal reference to run the motor at a constant speed. In cheaper machines this is generated by an RC oscillator, which is quite stable, but does drift (there is an internal trimpot for it) - in the really professional ones the servo reference is crystal-derived, and the speed is really rock-solid.
If you are intending to use your tapes on other machines you should try to make sure yours is running at the right speed - though it's not terribly likely anyone else's is, since no tape machines have been made for 20 years, and most of them have been poorly maintained. You should at least record a reference tone (A=440Hz is easy, then you can check it with a standard guitar tuner) so you do not end up having to tune the playback by ear when doing overdubs. If the 440 always plays back at the same frequency on your machine there is nothing to worry about - if it doesn't, that would indicate that the capstan belt is slipping or there is something wrong with the motor.
I occasionally send 2-track masters out for record cutting and always include the most accurate 1KHz I can - and instruct the mastering engineer to check it against theirs and vari-speed the machine if necessary.
Is there a calibration on the console that you can adjust? I would still trust consistent input from multiple sources as correct.