You’ll be glad to hear you’re doing nothing wrong. It’s quite correct that in words such as spare and inspire there is no noticeable puff as you release the /p/. The /s/ sound is itself a letting out of air; the /p/ simply closes this air off before you move into the vowel.
The Rule for pronouncing clear Stop consonants (p/b, t/d, k/g) says ‘AIM to release all Stop sounds evenly and equally with an audible puff of air’. This is an expedient instruction. That means, you follow the rule by aiming to release all Stops with a puff. Where your mouth cannot release the air at fluent speed, it won’t. The native English speaker’s mouth won’t do so either.
No /p/ air puff after /s/ is a perfect example of something that happens by default. But what about the other Stop where two lips close, the /b/? Does it behave the same way?
If you were to pronounce the two nonsense words ‘spee’ and ‘sbee’, a speaker with good aspirated ‘puffs’ will create a subtle difference between the two when speaking the words deliberately. Their attempt to release both Stops with a puff will allow you to hear a difference (the /b/ will sound duller than /p/). However, at fluent speed this difference will disappear and no air release will be noticeable after either Stop. This is no doubt the reason why English has no words spelt with both ‘sp…‘ and ‘sb…’ clusters; you could not tell two such words apart.
This is the theory but, fortunately, you don’t have to worry about actions that happen by default. Your instruction is to do what every English speaker ‘thinks’ they are doing i.e. releasing every Stop sound with a puff of air. By aiming to release the ‘sp…’ words with a puff, your mouth will create the precise, correct result.