Sorry. I don't think that a low performance comes from the antenna. A long wire as actually a quite good antenna if it is tuned to the right length and if you do correct impedance matching.

But as an RF engineer I notice several bad/strange things:

1.) No double sided PCB. Use a double sided PCB and let one side be a complete ground plane. This gives you a solid ground and stable impedances

2.) Are this serpentine traces a substitute for a proper coil? With "AM" you probably mean the medium wave range, like 500kHz to 1600kHz. PCB traces as coils or resonators can function in the 100s of MHz range but give very poor performance at these low frequencies.

3.) Not any tunable LC circuit - very strange in a radio circuit, especially in a transmitter. It is absolutely necessary to tune the impedance matching of the antenna to the output stage. Otherwise you will not get a good efficiency and range.

I also do not see any power transistor which could justify a 30V supply :-)

But if you really want a step up converter, why don't you use a chip designed for this instead of a microcontroller? With an internal regulation amplifier and proper phase/frequency response o f the control loop. Then you could adjust the voltage with single potentiometer and do not have to reprogram the thing. For your fixed duty cycle even e 555 timer would be enough.

The short answer for why I’ve built it the way I have is because I didn't have any RF experience when I made this. This radio was me trying to hack together something just to see if it work work.

To address each of your points specifically:

0. My “antenna” has not been cut to the right length or impedance matched. I just took the longest stretch of wire I had and connected it to my radio. I didn’t trim it to length because at the frequency where this operates (about 1MHz) I didn’t have nearly enough wire for even a quarter wavelength antenna. 

1. I used a single sided PCB just for simplicity. This is what I had on hand that was large enough. A ground plane does seem like a good idea.

2. I knew I was going to have a bad antenna, so I wanted to see if I made a long trace on my PCB if it could compare in performance to using a long and unbent wire (my version of an antenna). So no, the traces aren’t being used as an inductor, but the rationale behind them is definitely not good.

3. Being able to tune the radio and match impedances is absolutely something I would like to implement. I haven’t because I don’t know how to properly do this yet, but this topic has been on my to do list and is coming up. 

The 30V rail is just used to get the peak-to-peak voltage of the final signal to be a bit higher (about 15V instead of about 4Vpp as is found in the preceding stages of the radio). Little enough current is drawn that the TO-92 package transistor I used can handle the power dissipated. Using a dedicated chip for the boost converter would also be a good change.

I really appreciate your feedback because it’s nice to know what the biggest problems are with this project to someone who has RF experience. 

It's not "ugly", it has a very nice 3D printed case and the PCB looks nice also, it's just does not show much experience in RF design. :-) 

Impedance matching basically means that you transform the input impedance of your antenna to the output capability of your amplifier. E.g. if you have a 50 Ohm Antenna (around the value of a lambda/4 antenna) and we calculate with a power of 1W you have around 7V RMS and 141mA.

Now if you supply your output stage with 5V (what gives you your peak to peak maximum) you get at max. 1,78V RMS, so you need much more current for your power. You tried to do circumvent this with the high voltage.

In RF stages you can do a nice trick: You use an inductor or even a tank circuit in the collector circuit of the output transistor instead of a resistor, then your output voltage can swing above the supply rail. In this case you have to bring the transistor to a stable bias point: A resistor in the emitter circuit, bridged with a suitable capacitor and a defined voltage at the base.

If you use a tuned tank circuit, you get another nice byproduct: Your amplifier does not work for harmonics, which you do not want to transmit. They are filtered out and not amplified.

The other important aspect of impedance matching is the compensation of capacitive or inductive imbalances, which prevent efficient energy transfer from your amplifier to the antenna. E.g. if your antenna is too short, you can connect an inductor in series. This inductor has to carry some resonance current, so it should not be a serpentine trace but some good copper wire. You don't want to convert your precious RF energy to heat.

So: It's no surprise that your antenna is to short in this frequency range (wavelength of several 100m) but you can stretch it electrically and this "stretching" is part of the "impedance matching".

These are only some points of RF design. If you are really interested in RF circuits you should get some amateur radio books with circuits and theory.

There is way too much to write here :-)

If you need voltage converters, look for the TI Simple Switchers, like LM2596 (I think that could be a step down, but I am not sure, then the step up has a similar number) or similar. You get them even as complete modules from Aliexpress or ebay for a $. They are really easy to use.