Another option with an Aiyma class D integrated amp with vaccum tubes and a VU meter in addition to the tone controls, aux out, additional inputs including PC-USB), etc.. $109 after a coupon.
This would put you $6 over budget but get you a a more interesting and flexible amplifier:
https://www.amazon.com/AIYIMA-Bluetooth-Amplifier-Channel-Digital/dp/B09MLPC8X3
https://www.amazon.com/Passive-Bookshelf-Speakers-Tweeter-Renewed/dp/B08LTS9N67
The Emotiva B1+ on sale rn $220 or Refurbished JBL 530s rn $199. Both are awesome speakers and KILLER deals.
Agree the AD18 is a bit long in the tooth but will work. In addition to the excellent SMSL and Loxjie A30 mentioned previously, the Aiyima T9 ($110.50) or D03 ($110) may fill the bill.
https://www.amazon.com/AIYIMA-Bluetooth-Amplifier-Channel-Digital/dp/B09MLPC8X3
https://www.amazon.com/gp/product/B08CJZGT6H
https://www.amazon.com/gp/product/B07YZDS9LF
https://www.amazon.com/gp/product/B009GUTJ34
Aiyima A07 / "300w" x 2 class D amp with line level output (for the sub)
Neumi BS5 passive bookshelves (5" 2-way)
Monoprice 60 watt 8" powered subwoofer
$246.47 with free shipping after the $20 instant discount coupon on the Aiyima.
Having tone control could be nice but the Neumi's are fairly neutral sounding speakers particularly if you plug the ports which should be fine with the sub filling in the low end.
Otherwise if you can stretch by $6 you can remix this configuration if you are OK with the amazon Renewed version of the speakers, and get an upgraded amp with tone controls, a VU meter, bluetooth, PC USB + additional digital audio inputs, and vacuum tubes on the preamp side:The list below would put you $6 over budget but get you a more full featured amplifier:
https://www.amazon.com/AIYIMA-Bluetooth-Amplifier-Channel-Digital/dp/B09MLPC8X3
https://www.amazon.com/Passive-Bookshelf-Speakers-Tweeter-Renewed/dp/B08LTS9N67
No direct experience with these products , but this is a decent baseline for a system, price-wise:
https://www.amazon.com/dp/B099DX4RY3/ and https://www.amazon.com/dp/B09MLPC8X3/ lets you do a PC via USB, CD via coaxial, phone via Bluetooth, and you still have 2 inputs left. Tons of options for speakers, these are on sale right now. https://electronics.woot.com/offers/jamo-s-803-bookshelf-speakers-pair-1?ref=w_cnt_wp_0_77