I live in a developing country with a ton of bribery that is way,
way down the Transparency International rankings.
Your ideas come across as pretty naive and out of touch with how bribery works in the real world. Which is okay, not everyone is an expert on everything. But your comment would come across a lot better if you had used some tentative language instead of definitive language on a topic it is clear you don't know much about in the real world.
Bribes are rarely solicited in the real world, so that's the first place you theory falls down. I've only rarely had someone say something that could even vaguely be construed as soliciting a bribe. They aren't dumb, after all. Instead what usually happens is you get a run-around for unspecified reasons until you offer the bribe. You submit the paperwork and they tell you it is filled in incorrectly but don't tell you what part is wrong or how to fill it in correctly. The nurse only checks on you once a day until you give her an envelope of cash. The accountants auditing your books ask for receipts of every single everything until take them to dinner and offer them a present for all the hard work they've done so far. The fire department finds all kinds of ways to not certify your new office for occupancy until you give them some "coffee money".
The second problem with you theory of bribery is the offering bribes changes the structure of compensation dramatically. The expectation of receiving bribes becomes part of the expected total compensation for a position. But this gray money -- the delta between official compensation and bribes -- becomes a revenue stream that everyone above you on the hierarchy can and will tap into.
In practice this means you have to pay a huge bribe to get any job. Where I live, to get a job as a police officer, or a nurse, or a flight attendant, you have to pay a bribe to whoever is hiring. Not a small bribe. Like a year of salary. Which means either you come from a family rich enough to have that kind of money laying around or borrow money from a money-lender.
How do you get the money to pay back the money-lender? By taking bribes, of course.
Your boss knows approximately how much you're making in bribes each month and will expect you to pass it along every month. After all, he borrowed a ton of money to pay a bribe to get his promotion and needs to pay off that debt.
And if you're complaining about >10% of the total project cost going to bureaucratic processes, your idea would result in 30% or more of the total project cost going to bribery. Since that's exactly what happens here.