You should do proper wodu to be in the purity state (Tahara) in order to perform your prayers. Sharee'a is dynamic; if it's difficult to wash your feet in public, you can wipe over your socks (under some conditions) once with your wet hand and your wodu' will still be proper. Prophet Muhammad did that, as well as his followers.
This is a complete fatwa about the conditions, they're easy anyway, I can summaries as:
- Make wodu before putting on your socks
- The socks must be pure (Tahir)
- Wiping over your socks if you are doing regular wodu, not after janaba
- Wiping is permitted for a single night/day (3 in case of travel). Afterwards, you can take them off, do wodu and wash your feet, put your socks back, restart counting a new period if you wish. So your feet will be washed at least once a day.
Notice the following scenario: Make wodu' and wash your feet - put on socks - go to the toilet - Make wodu' and wipe over your socks .. you can then pray normally. At this point, if you took off your socks, your wodu' is over! Even if you did not break it. So your socks should be on to preserve your wodu' when you're out of home.
PS: I normally make wodu' when i get up in the morning, wear my socks, goto work, wipe over my socks for the next four prayers :) It's an everyday scenario.
Another PS: Please dont say "do what we can and leave the rest for God". If you are (really) unable to do it, then Allah would not have made it a command. Islam is easy and dynamic, just ask how to do it.