> Don't have a clue how wide the range is
Roughly third-root of 350K/100K. 1.51:1. Just enuff to cover 20% tolerances.
Say 360Hz-540Hz.
Taking the upper resistance far higher than the others (270K) won't drop the frequency much (the third-root approximation is failing).
For wider range you may use a 3-gang pot. Now freq varies with resistance. 1Meg to 100K will approach 10:1 freq range.
You can switch caps for other ranges. A 3P3T switch will manage say 20-200, 200-2K, 2K-20K. Maybe. The caps get very small above 10KHz and will interact with the triode's grid capacitance.
Multivibrators are dandy for square-waves.
There are ramp-waves on the grids of a symmetrical multivibrator but extracting them loads the timing action and spoils it.
There are tons of ramp-wave tube circuits for TV sweep, much harder to get a symmetrical triangle.
For a wide-range Sine: don't be foolish. eBay yerself an H-P 200AB. Don't worry about condition, age, or known-working. They are easier to fix than the sick Champ in your closet. The only sticky-spot is that the massive tuning-knob's shaft grease tends to turn to gum in only 100,000 hours of always-on duty. And once gummed-stuck it is hard to get at the shaft-screws to dismantle it. Eyedropper some nasty solvent into the bushings and be patient. When it will come undone, scrub the shafts and lube with good engine oil (3 drops from the car dipstick, right after an oil-change, is ample). Do it again before the year 2029 rolls around.
For square/triangle/"sine", you just can NOT beat the little chips made for the purpose.
Voltage-controlled tube oscillator..... they exist. Any oscillator will drift a bit when supply voltage changes. Some GHz tubes can only be tuned with an electrode voltage. FM radio AFC uses a second triode as a vari-gain capacitor to drift the LO. Conceptually a multivibrator can be VCOed, and this is easy with transistors, maybe not so easy with tubes.