Daf Yomi just started פרק כירה which discusses these issues.
The first mishna there has a gezera that you are not allowed to put food even on a covered fire on shabbos (even where there is no problem of bishul). The Rishonim offer 2 reasons why:
1. שמא יחתה בגחלים - maybe you will stoke the coals
2. מחזי כמבשל - it looks like you are cooking
Based on this a number of poskim (R' Moshe, R' Eider, R' Willig) hold that the issur doesn't apply to a non-adjustable hotplate because there is no שמא יחתה בגחלים because you cannot change the temperature, and it is not מחזי כמבשל because no one cooks on such a hotplate. Therefore they hold that you can heat up solid cooked foods on shabbos on a hotplate. The Shmiras Shabbos disagrees. Note, it is not so simple to just be machmir, because you are taking away from your oneg shabbos, it is clearly more oneg shabbos to have hot food.
R' Moshe has another interesting chidush (not למעשה). He says that our stove tops should never need a blech. He explains that the whole reason for a blech is to prevent שמא יחתה בגחלים, with out stove tops that is impossible, there are no coals to stoke. What about turning up the gas? That is adding fuel which the gemara was never gozer on.
Interestingly enough, R' Aharon Kotler also dealt with this point albeit differently. He said that since today the worry is that a person will turn up the gas, the blech needs to cover the knobs (maybe it doesn't even need to cover the fire) so that the person will have a reminder not to turn up the gas.
Lastly, the Chazon Ish holds that the idea of grufa uktuma in the Mishna is that it lowered the heat significantly and therefore he holds that what we call a blech is no good as it doesn't lower the heat. According to him you need a blech like an upside down pot which does lower the heat.