YES YES AND YES!!!!
The situation you describe is a pain in the proverbial every working day for crews of TCAS equipped aircraft. Transponders without altitude information should be banned.
All airline aircraft operating in UK airspace are now required to have TCAS fitted. TCAS is a wonderful system and works extremely well, but its whole principle is based on the idea that other traffic will have either (1) A transponder which provides altitude information; or, (2) No transponder at all.
TCAS does not work at all well when there are aircraft with transponders which do not provide altitude information. This can happen for two reasons; (1) The aircraft does not have an altitude encoder fitted; or, (2) The pilot has not switched the transponder altitude information on.
The real rub comes when a TCAS equipped aircraft is operating in controlled airspace which has non-controlled airspace below it.
For example, the TCAS equipped aircraft might be in cloud at FL90 and the lower limit of controlled airspace might be FL50. Flying below the lower limit of the airway is an aircraft with a transponder but which is not providing any altitude information.
To the TCAS equipped aircraft, the other aircraft might appear at 12 o'clock, opposite direction, one mile, with no altitude info. Although the two aircraft are in reality well separated vertically, the TCAS system will assume that the other traffic might be at the same level - but of course it has no way of knowing for certain.
Unable to generate a resolution advisory, the TCAS system can only generate a traffic advisory and will announce, "TRAFFIC, TRAFFIC" over the audio system and will present the crew with a big yellow blob on the TCAS display, representing the target.
The hapless crew then has to decide how to deal with this information. There is no easy answer to this. Whilst dealing with this dilemma is no big deal in the cruise, in busy airspace during an approach to land with multiple radio frequency / heading / altitude / speed changes and checklists to deal with these TAs can be an unnecessary distraction. On one approach into London I had seven unnecessary TAs on traffic with no altitude information within 10 minutes.
All the while the non-TCAS aircraft has no idea of the consternation he is causing those around him.
Please please please, owners and operators of aircraft with transponders :
If your aircraft has an altitude encoder please turn it on! If you don’t know how to turn it on, please ask someone. Most GA transponders have a position marked "ALT" which is the setting required. The position marked "ON" does not provide altitude information from most GA transponders.
If your aircraft does not have an altitude encoder and you don’t want to get one, please consider turning off your transponder unless it is absolutely needed. Alternatively, I have seen altitude encoders advertised for about £100. You might consider fitting one of these to your aircraft as it would be of mutual benefit.