I'm on mobile with really slow data right now so I hope someone else can go into more detail but REAL QUICK
A thermostat is vital and the most important equipment because it regulates your heat sources, preventing it from overheating or burning your snake. Trust us when we say snake burns are nasty business and the vet bill will cost way more than a thermostat. ALL heat sources must be regulated. This includes your UTH AND lamp/CHE
Keep in mind what feels warm to you, a mammal with a natural body temperature around 98F will feel significantly different to a cold bloded animal.
The only way to take ACCURATE surface temperatures is with a temp gun not a thermometer which is best to measure AMBIENT temperatures. Chances are your temps are too warm since they are unregulated. Make note you are to measure temps from your heatmat UNDER the substrate against the glass directly. BP's can burrow.
My phone is driving my crazy but hoping this quick reply will help give you an idea that yes, a thermostat is important. Its not optional, You need one.