serious question: do you have any concept why commercial manufacturers do their soldering station power circuit using mosfets?
I don't know why some commercial soldering iron stations use MOSFETs. There are probably various reasons, such as availability, being easier to control with a microcontroller, without an opto-isolater and being able to switch on and off, without latching, but I doubt efficiency is one of them. Look at the data sheet for a typical bridge rectifier, taking note of the full load voltage drop, now compare it with a TRIAC.
TRIAC
https://media.digikey.com/pdf/Data%20Sheets/NXP%20PDFs/BT138%20Series.pdfOn state voltage, 1.65V, at 15A.
Bridge rectifier
https://docs-emea.rs-online.com/webdocs/0fbb/0900766b80fbb6ae.pdf1V per diode, at 20A, a little less at 15A, perhaps 0.9V, but there are two diodes in series, so that's a total of 1.8V!
Note, that I'm not even comparing like with like: on the data sheets linked above, bridge rectifier is rated to 35A and the TRIAC rated to 12ARMS. If I compared a bridge rectifier and TRIAC, of the same current rating, the bridge rectifier will look even worse.
The losses in the bridge rectifier will dominate the MOSFET losses and will be higher than the TRIAC losses, for the same load.
Also note, that it makes no difference, whether the AC is rectified and the element operated with unfiltered DC, or the element is in series with the bridge, with the MOSFET directly on the output: the losses due to the rectifier will be the same.
If you want low losses, then use back-to-back MOSFETs and a photovoltaic opto-coupler, similar to the one I linked to above.
If you want an easy life, run the element from unfiltered DC and the MCU from a regulated supply, consisting of a diode connected to the bridge rectifier, a smoothing capacitor and the LM7805. Use a MOSFET to switch the element, but don't worry about using one with a really low on resistance, so long as it's good enough not to require a big heat sink.