How to split x items/s from a blocked full belt?

Assuming you are okay with an approximate rate:

Use one or more inserters to set the desired rate of item transfer. (The best way to do this will depend on how much inserter stack size research you have done, but once you have a satisfactory setup you can keep it working by using the "Override stack size" option on each inserter.)

(A splitter splits off a short belt which two fast inserters unload onto a third belt.)

In this setup, I've used a splitter to make a short terminated belt so that the inserters can grab more stationary items, for more consistent timing and lower power consumption.

You can connect two belt tracks with green/red wire like this:

====S===== (blocked)
    SAB=== (10 items/s)

= - belt
S - splitter
A and B - belts connected by wire

Set B to read belt content on hold.

Set A to enable/disable when amount of items is below some value. You'll have to play around with values to get exactly 10 per second.

More information about circuit network can be found here:

Just use a splitter with output priority set to where you normally want it to go, then when it blocks it'll start to "spill" onto the other belt. No need for counting or other complex combinator setups.