Why do non-native English speakers get the present participle wrong?

So, in traditional grammar these cases would be considered gerunds, not present participles, because they head noun phrases. Modern grammatical analyses of English (such as the Cambridge Grammar of the English Language) analyse gerunds and present participles as a single construct called the gerund-participle.

In any case, this error is common because some languages (I am personally familiar with Spanish, but probably there are others) have parallel constructions which are identical except for the use of the infinitive instead of the gerund.

For example the English sentence

[A new infusion [of cash] ] [allows [making the film] ].

translates as

[Una nueva inyección [de dinero] ] [permite [hacer la película] ].

The Spanish word hacer is the infinitive form of make, and, for native speakers of Spanish, this error is in fact quite common.

Uhm, could you please try giving a better example? Both the sentences you have posted as an example sound a bit awkward, I would almost say they are broken.

Remember that "it allows one to make a film" is correct.

Also, "it allows making a film" is correct.

I know this isn't exactly what you wanted to know, but unless you provide a concrete, proper example, I think it's very difficult to distinguish what is really being asked here.

P.S.: It should also be noted that unless I'm missing the context, your sentences should start with with the new infusion of cash.

EDIT to reflect your edit:

All those examples you have added seem to be perfectly fine to me. There are even cases where your theory would fail, imagine:

They allowed me to do the job.

It would be incorrect to say the following:

They allowed me doing the job.

On the other hand and taken from your example, imagine:

This application allows you to find your cell phone.

Do you think this statement is wrong? So how would you write it? Like this?

This application allows you finding your cell phone.

Because this is obviously wrong.

Which comes to a conclusion:

Allows you to find* implies that you will be able to actually find the phone after looking for it. It talks about the successful action of finding.

Allows you looking for would imply that the application helps you with the looking process, but it isn't necessarily everything you need to succeed in your search.