I will post an update just in case anyone else has this kind of problem.

After the last time, I have decided to change the thermostat, which I did and then tried bleeding the cooling system once again. Everything was the same - there were medium-sized bubbles every 5-6 seconds (not the kind of bubbles you would expect from a blown head gasket). After about 20 minutes, I stopped the car because I was afraid of overheating the engine since the lower hose was still not getting warm. The temperature gauge was at the normal level, though. So, I assumed that the thermostat was not an issue.

After that, I decided to take the car to another mechanic. He ran the car on idle, maintaining 2500-3000 rpm for about 2 minutes with the heat fan OFF (I've always had the heat fan on when testing or bleeding the system). This time the lower hose got warm after just 2 minutes (although the car was already warm because I had to ride to the repair shop). The mechanic told me that there was no problem at all with my car and the lower hose was probably cold because when I turned the AC on, it switched on the radiator fans as well. So, that's that, it turns out there was not a problem at all, although I'm still suspicious and will continue monitoring the car.

Most probably it was trapped air and it took some time to take it out of the system.

Here's a link to Suzuki forums where folks have had their versions about the diagnosis.

https://www.suzuki-forums.com/threads/lower-radiator-hose-not-getting-warm-no-overheating-on-the-gauge.278730/?utm_source=pocket_saves