Nothing will get hot. It doesn't work that way.
Okay look, the power supply is first of all a transformer. 120V AC comes in, goes round and round the primary windings of the step down transformer that drops it down to (in this case) 15V AC. Always AC. Transformers work on the principle of induction. The rising and falling magnetic field around the wire induces a current in the secondary winding. Same reason we don't want power cords, speaker cables, etc coiled up. Induction.
Okay so the ratio of primary to secondary windings is what determines the step up or down in voltage. The operative term being: voltage. Hate to get all mathy but V=IR. Or I=V/R. However you want to write it. Point is voltage is proportional to current. In other words you can step the voltage down and have more current, or you can step the voltage up and have less current, but you cannot use a transformer to increase both voltage and current.
Okay so we had a 15A circuit and we stepped the voltage down from 120V to 15V so we should be able to get 100A easy, right? Not so fast! Look at how small that dinky little transformer is. Look at how hair thin the wires are. (Seriously. Open it up. Have a look inside. Live a little.)
Induction simply is not gonna deliver that kind of current through that thin a wire with that small a set of windings. There's a reason your neighborhood power transformers are as big as they are. Have to be to deliver that much current.
So the wall wart transformer is very low amperage. The 15V comes out the secondary and then goes through a set of rectifier diodes. Diodes allow current to pass one direction only. That's how we get DC. At this point the power is notchy because of the diode switching so the next thing is one or more power supply caps. Their job is to smooth the ripples from the diodes, and provide reserve power for surges and peaks. Look inside any component, they are all the same: transformer, diodes, caps. Just like that.
Where is the part that is gonna generate any heat? Transformers generate a tiny amount. Diodes a whole lot more. Some need heat sinks. But they are sized to the power needs and so run very stable. Its the transformer that limits the whole thing.
Just to give some idea what I'm talking about, I forgot that my modded conditioner had one outlet intended for only a CDP and it used a little dinky 20 watt isolation transformer for the CDP. I plugged TWO Dayton SA1000 sub amps into it. Everything ran fine. Well except for the subs clipping at lower volume than I thought they should. But it was very high volume and it was only when I opened the conditioner to do another mod that I noticed my mistake. Point is that dinky little transformer ran two monster sub amps with no heat no problem. Because it was sufficient for the average draw, because the Daytons have their own big power supplies with lots of capacitance. Understand?
The problem in your situation is neither your turntable motor nor your phono stage has that kind of built-in power supply. They both are relying on the connected DC power supply.
So now you know what to do, right? You can run it and see what happens. Probably no problem. You can measure voltage drop like the other suggested, if you want to get all mathematical about it. You can get another better power supply. Or you can add capacitance. A bigger better cap and you will be good to go.
These things are not that hard. You can learn a lot playing around with this stuff. Diodes that will make a big improvement in SQ are not that expensive (under $10 a lot of them) and even a really good cap might still be less than you would pay for a complete power supply. You might even scavenge one out of some other gear. Cap values affect frequency response in crossovers but in power supplies its straight up more is better. So why not try? Go ahead. Live a little!
