To be clear, one cannot estimate! In this case, you can use ANY count that an official has once the ball gains inbounds status to alter the clock (10-sec or five second or if an official notices the clock isn't running and begins to count) and sum them all, but what you may not do is subtract time from the clock in the absence of any count.
You may not like it, but the game must be timed according to the rules, not according to what makes logical sense.
Officials should not be expected to perform the duties of either the timer or scorer. Mistakes happen and those people screw up from time to time. People need to learn that the officials can only fix so much. The rest they have to accept and move on.
Btw in the case of a team being out of TOs, I would definitely start a count on the inbounds so that a minimum of one second could be corrected and the technical foul avoid BY RULE. What I won't do is fail to adhere to the rules.

You may not need exact knowledge, but you need to have some definite "at least this much" knowledge. Chances are, in this situation, you've at least had 1 second of a 5 second count, but if you haven't, then you can't take any time off by rule. Now, nothing says you have to have a 5 second count to be counting.

Otherwise, if you say "why not just take one second off", others will say "Well, I know it was at least 5 seconds". I'm not much for slippery slope arguments, but it seems the rule is the way it is for a reason.

So what if the in bounds pass goes into the front court and A takes 4 seconds after having the ball in their possession to start running a play, then B decided so closely guard and then You have a count to 4 and then realize the clock hasn't started. Could you just take off the four seconds for the closely guarded but do nothing about the other four seconds as no one was actually counting anything?

Yes, because I have no idea how long B took standing there without playing defense unless I had a count. By rule, you need definite knowledge. In your case, I have definite knowledge 4 seconds should come off.

Oftentimes people forget that those correctable errors refer to errors made by officials. Any error made by scorers/timers can be corrected.

Agreed. "Have" a count of something. Team A won't complain because they know that some time should come off the clock. Team B won't complain because some time did come off the clock.

In your head, keep a count. Even do an arm swing if you wish. ;)

The officials who do this are also the officials who would have noticed the clock not running in the first place, and would correct the error without notification from the coach.

+∞

My point was not that this isn't something that can't be corrected, but rather that this error shouldn't cost a team a time out. As officials we need to know the status of the clock and we need to be in the habit on making sure that the clock runs when it is supposed to and stops when it should.

And you are incorrect about that. In post #13 of this thread I provided the rule for you and it clearly includes a timing mistake.

Thus if a coach wants to contend that a timing error occurred and no correction can be made, then BY RULE it will cost his team a time-out.

I would think that the spirit and intent of that rule would result in no timeout if the crew confirmed a mistake was made but didn't have any sort of definite count to make a correction. In this case, a correction can be made if an official has a count. Whether the team is charged a T or not shouldn't depend on whether an official had a count or not.

I agree, if it is acknowledged that a mistake occurred, but no definite knowledge of how much time should come off, then the team shouldn't be penalized. The coach met his burden of pointing a real timing mistake.

And I do have at least one supervisor who tells us we need to take off at least 0.4 if a player on the court catches ball, based on the rule regarding needing at least 0.4 on the clock to catch and shoot.

+1. My sentiments exactly.

That seems a bit unreasonable since catch and shoot is two actions with the "shoot" part of it probably taking longer than the "catch" part of it. Should something come off, yes, but I'd suggest that the minimum time to catch the ball is less than the minimum time to catch and shoot.

This is how my association interprets this. I was taught that in a situation like this (where it is definately not zero or close to zero), briefly get together with your partner(s) if necessary and start asking yourself ... "Was it definately at least one second (yes); was it definately at least two seconds (yes) ... was it definatly at least five seconds (not sure) -- Subtract four seconds off the clock as you have definite knowledge that at least four seconds have elapsed.