The cabin heat will come on no matter what OAT is. If OAT higher or lower than cabin temp, it will change recirculating or fresh outside air setting automatically once fan turned on. Turning on fan powers up the HVAC system.
Make sure both front and rear heaters are set to full HOT and turn fan on HIGH for at least minute, as you drive or run engine for 10 minutes or longer once at operating temp. Again, turning on the fan powers up the HVAC systems, which will then accepts and adjust to the setting you have set. You can turn fan back off, but keep heat setting on HIGH front and rear (make sure rear heat control set to hot). You've not said what year or if a NAV or no NAV system. But I do treat both the same in that I turn fan on, to accept settings like temperature or fresh air.
Heater core failure or plug is not common in a series 100 that is properly maintain. If tap water is ever added it, this will increase likelihood of a plug or failure of cores. There're many types of coolant today, adding the wrong coolant can create a plug or jell the coolant.
You mentioned burping "around 30 minutes each time" not sure what you mean. If you mean; Just top and drive for 30 minutes, then check again next morning (8 hour cool down) that's good. The system is actually self brupping and draws coolant from reservoir as engine cools down. All we are really doing is making sure it's topped with coolant so system can do it's job. It expels coolant into reservoir as coolant expands (heats). If reservoir runs dry it may suck air into top of radiator during cool down. If air in top of radiator, then this cycling in and out of reservoir may not function properly, depending on how much air. This is why we check radiator level first. If radiator cap bad, this moving in and out of reservoir doesn't work as designed either. Old cap should be replaced. You can test a cap, but good luck finding anyone with a cap tester.
Keep burping each morning. Once your radiator is found to the top of the neck/cap in the morning (after a cool down), it usually only takes one more mornings of adding to reservoir before level holds from then on. I've only had and air bubble get stuck in a heater core once. It was after I blew out heater coolant lines with air. This once, I serendipitously stomped on the brakes WITH ENGINE AT OPERATING TEMP and air bubble released. Front Cabin heat begaim working a minute later. I did then need to start burping/topping procedure all over again.
Coolant is very sensitive to temperature. So on a cold morning with OAT of say 40F, the level in reservoir will be lower than when OAT is "say" 80 F. Even 10F will make a noticeable difference in reservoir level. For this reason I set reservoir at low line when OAT is -10 or -20 F (Whatever expected annual low for your area). But if OAT is 80 or 90F, I'll set level at halfway or even higher. When engine is at operating temp, I like to see reservoir at high mark and feel upper radiator hose hot and full/hard.
Minor coolant leaks will not always show on garage floor. Most leaks are just when engine hot and running. But if using Toyota red it will leave a pinkish dry flake (when dry) trail behind from leak.
A weak thermostat opening to much or too soon, engine will run cool and so will cabin heat. But if front cabin heat cool and rear very hot something wrong. Our cabin heat blows very hot, especially the front.
You never said why you wanted to backflush in first place. Was something not working or was this just a 2yr or 30K PM coolant flush?
Did you have heater tees hoses off and possible reconnect hoses wrong. (not easy to do, but possible? Picture are helpful)
Once coolant level set, level only varies depending on temperature. So continually adding indicates a leak. We can talk more about leaks and how to find if your sure you have one.
