How to replace an alternator on a silplicity

Sounds like you have a charging problem and most likely have a bad alternator. Charge your battery and then install it (only if you removed it to charge of course). Once installed, check the vintage with a meter and it'll be 12 Volts or Very Close, then start it up and check the voltage reading at the battery again, if it doesn't say about 2 Volts more it's not charging and the alternator is bad. It should read about 13.5-14 Volts when running and the alternator is working properly.

The battery is only there to start the tractor(or vehicle, etc.), once it's running the alternator takes over and powers everything while maintains the level of charge in the battery so that it'll be ready to start your tractor back up when you cut it off once done with it. When the alternator is bad, it'll run until the battery is dead so you're Limiter to whatever the amount of Volts are in the battery and once it's drained it'll just shut down.

If your tractor allows you to do this comfortably & safely there's a simpler way to verify if the alternator is working properly, this works on anything that has a battery & alternator so cars, trucks, equipment, etc:

Plug the battery up and leave one terminal loose enough so that you can twist it off by hand, start it up, then pull the battery terminal off(either one). If it cuts off, the alternator is bad. If it stays running you've got another issue. But that's rare when anything has the symptoms you're dealing with.

Either method works fine, but if you don't have a volt meter nearby then you can always pull a battery terminal off while running and check to see if the alternator is charging as it should be.

All riding mowers that have a battery start have a charging system, per se, that runs the tractor after starting and provides charging power back to the battery. If this wasn't the case batteries would need replacement about once a month, unless you placed them on chargers between use. That said, the don't use the type of alternators that cars do (at least not the smaller tractors), they use stator-based alternators. Here is a sample of what one looks like schematically (from a Kohler-engined Cub Cadet) (yours may differ slightly).

have battery and alternator tested your battery may not be holding a charge or alternator damaged by jumping tractor

Charging control system has failed and is overcharging the battery. Probably the alternator diode, not sure if it's a replacement part on this model, you may need to replace the complete unit.

charging system need to be check... alt. some of these tractors have the charging reg. inside alt some have one mounted outside follow wire the red one should be one big (thick) one and a thinner (smaller one) this small one is the charging circuit. if it goes from battery to alt. then regular is inside alt.

If you hold a wrench against the back of most alternators when it's running the wrench should be attracted to the alternator which would indicate that its charging. If it isnt' charging then the regulator inside the alternator is burnt (typically) if it is charging then check the wires to & from the alternator.

You need a new alternator...Alternators regulate current to the battery to keep it charged.

The wrong seat switch was installed. There are black ones and gray ones. I believe you need the gray one for open circuits.

the starter puts out the amps to charge the battery, you have do an amperage check to find out what its puting
any good parts store will set you up with an inexpensive tester

I don't see how this is related to the float?
With the engine running, measure output voltage from the alternator/charger at the battery. It should be about 13.5 vDC. It should increase a little bit with higher revs. If voltage doesn't change with engine running vs. engine off then you probably have a bad alternator which probablly has to be replaced or repaired by a service shop.