Are you sure the PSU is the culprit? If you press the BIST button does the green LED light and the fans turn on? As for a case swap, the SFF versions are near impossible to swap to another case. The best option would be to purchase a compatible Socket 1151 MB from eBay and move all the components on to that MB in another case. This YouTube video offers some insight into the process.

3070 sff motherboard has proprietary form factor which does not fit any non-dell case. Similarly the Optiplex case is also proprietary made for oem board only and does not work w standard motherboard.

3070sff motherboard is c/w Dell oem TFX psu that has 6+4 white pin.  there is a long list of compatible psu including those for 3060/3050, 7060/7050, 5060/5050 SFF/MT.

How to run self-test without the button on the PSU

1. Turn off the computer.
2. Disconnect the power cable from the PSU, and wait for 15 seconds.
3. After 15 seconds, connect the power cable to the computer.
4. The LED light turns on for 3 seconds and then turns off.
   1. If the BIST LED turns on, the power supply is delivering power to the computer.
   2. If the BIST LED does not turn on or the PSU fan does not spin correctly, the power supply is not delivering power to the computer. 

3070 sff psu: 200W typical 85% Efficient PSU (80 PLUS Bronze); ENERGY STAR complaint, Active PFC

You can also use 3060/3050 etc. 240w psu that has 6+4 pin which is c/w 3070 sff board.

6F0T1 240W Acbel
22M8Y 240W Acbel
D5PCJ 240W Bestec
W2DN4 240W Bestec
4GJV9 240W Chicony
J61WF 240W Chicony
5WNDJ 240W Delta
CJG5K 240W Delta
F484X 240W Huntkey
DK87P 240W Liteon
JVW1K 240W Liteon
R0NM7 240W Liteon

there is no button for psu it turns green for a few seconds then goes none

it doesn't have BIST button

Re: there is no button for psu it turns green for a few seconds then goes none

according to Dell if the LED light turns on for 3 seconds and then turns off, the power supply is delivering power to the computer.

PSU Tester is also an idea; many on Amazon.  Were any modifications made since the PC was purchased or recently?

Something redxps often recommends also:  How to Perform a BIOS or CMOS Reset and Clear the NVRAM on Dell Computers  And use a new size 2032 battery when doing this.  No guarantees, but can be a first simple step in troubleshooting.  I'd be remiss to not mention.

It seems that after a few hours of not trying to power on my pc and as of now, the pc is powered on. I just noticed that the power button light is not lit.
Some of the steps that I did was cleaning the pc, replacing the thermal paste, and changing the CMOS battery, but after doing the steps it didn't work out. 

It sounds like you may have Googled the error code - smart.  : )

Did you already try reseating the front I/O cable?

Power switch will come with new cable.  May also be available on eBay.  https://www.gotlaptopparts.com/products/dell-optiplex-3070-sff-genuine-desktop-power-button-cable-6m58d?utm_source=shopsheriff&utm_medium=amp&utm_content=link-to-full-site&_gl=1*1l40zny*_ga*WWxJdUE2ejBtdW1zQV90OVBjdjUzN0tENGVfM3JUYnNub3FKa0tMRWpVOUFYRHJjOEMxMFE5Sm0xNjNrb21YRQ..