Virg, Jim has long experience working with tractors of your vintage, I would check out the voltage sensor as he suggests if the tests I share with you fail to find the cause of your problem. The component he mentions is shown in the diagram below copied from the Wiring Diagrams PDF found on this site (see Main Menu).
The wiring diagram shows only a single wire for the alternator. It does now show a regulator, but that doesn't mean your engine doesn't have one, I know the 16 HP Briggs & Stratton engine I once had in my 782 had a regulator attached to the outside of the flywheel housing. According to their own documentation, Briggs & Stratton equipped engines of the era with no fewer than seven (7) different alternators and two different regulators. One of the alternators actually put out only DC voltage (there appears to have been a diode placed in the line out), of the remaining six (6) types, three (3) had AC two-wire outputs and three (3) had single-wire outputs. Yours is apparently one of the three single-wire units which (I think) came in a 5 Amp, a "Tri-circuit" (two-wire output from the alternator yoked to form a single wire output to the tractor), and a 9 Amp version. I suspect that if your output is genuinely a single wire AC output going into a square box with a screw in the middle, your output may be limited to 5 Amps, but is more likely 9 amps; if you have a two-wire output and the regulator has two screws, one on each side, then you may have a 10 Amp or even a 16 Amp output. The Briggs document tells us that, "The proper flywheel part number and/or the alternator magnet size will determine the alternator type or output. See repair instruction manual for additional information." I know that I could not increase my output to the maximum 16 Amps without buying a new $200.00 flywheel, but I could upgrade my alternator and regulator to a two-wire version and go from 5 Amp output with small magnets to 10 Amp output still using the smaller magnets.
Anyway, if you want to verify the other components in the system, beyond the voltage sensor, then you will want to check the AC voltage being output from the alternator BEFORE it goes into the regulator, and then check the level of the DC voltage coming out of the regulator (which is most likely nothing much more than a bridge rectifier in the case of the smaller, square, unit). You should be well above 24 volts AC at full throttle conditions (3200 to 3400 RPM) into the regulator and over 13 volts DC out of the regulator with the motor running. You should try to get the readings with everything connected, but given the nature of the Briggs connectors, you may have to break the line to get one or the other of them. Put the black lead on ground in each case if you only have one wire out of the alternator. If you have two wires, get the AC out by putting a lead on either line, NOT to ground; your DC reading will still reference ground, though.
Alternatively, if you put a volt meter from ground to the battery before you start the engine and then again after it is running, you should get an idea if the system is healthy. You should get in the neighborhood of 12 volts before starting, and you should get in the neighborhood of 14 volts after it is running without the PTO or lights on. If the charging system is working OK then all you have to do is swap out the voltage sensor as Jim suggests, or junk it and mount an ammeter or voltmeter on the dash (if you can find the room).
If you suspect there is a problem with the charging system, the following are known issues often revealed upon further inspection. (1) short to ground (anywhere), (2) poor connection (anywhere), (3) bad regulator, (4) bad alternator. The alternator can fail because of (a) wire insulation damage from rubbing on the sheet metal or frame into the regulator, (b) because of wires in the windings shorting to each other or to ground, (c) or because the magnets have come loose on the inside of the flywheel and are all stuck to the stator (as was the case with a Kohler Mag 18 I picked up to replace the Briggs motor when it failed). The magnets are actually the easiest problem to fix, all you need to do is buy some epoxy and glue them back on in precisely the order, orientation, and spacing as they were originally and the flywheel works good as new. I used a drill bit to space them all out evenly. Be very careful to preserve the order/orientation upon removal if you find they have come loose. I know that my tractor performed just as you describe, there was apparently enough juice produced to keep the battery healthy enough to crank the engine, but not enough to keep the needle above zero when I used the PTO, and if I turned my lights on while mowing, it would kill the battery.
Hopefully, all you will need to do is scratch some rust off and re-attach your voltage sensor and all will be well.
Good luck, and again, welcome to the Forum.