The holes are called "VIAS" and they do interconnect the tops and bottoms AND the middles (if you have multi-layer boards) If you have a connector where the pins pass all the way through the board and out the other side, then the pins will carry the load to the opposite side - but you have to use a lot of solder to fill up around the pins to get from the one side to the other and let it all heat up or you will not make a good connection. From what I see, the pad under the "I" of INPUT looks pretty bad. I don't envy the work ahead of you. You could use direct jumpers but I assume you have a connector that is supposed to go here?

For what it is worth, I have done this kind of work for over 30+ years.

Honestly, unless you have done this before, I highly recommend you not attempt this repair. You can wreck this board in a heartbeat if you overheat the pads. Do you know anyone good with a soldering iron and is skilled in this kind of repair?

Can you show me what the back side of this connection looks like? I may be able to walk you through a repair.

A stitch is probably your best solution. You are in luck as this looks like a two-layer board (Top and Bottom copper only). You may need to scrape off some of the green coating leading away from the pads if you suspect they are bad and solder a wire over to a good, clean part of the copper trace. Good luck. It is possible.