Brexiteers require a deal to be formulated that will gain enough support placating the DUP is an important part of this as they are unlikely to receive much support form Labour or the remain supporters in parliament needed to get a majority. It's worth remembering the hard brexiteers primarily want brexit they are simply OK or happy with the UK leaving on harsher terms as long as brexit is delivered.
The issue of the Irish border is perhaps one of the most important issues for any brexit deal as it it isn't a question about some benefit that might be received at some point in the future but something that handled poorly could lead to civil disturbances in IslandIreland that might even result in incidents in other areas of the UK and potentially island ceding from the UK, this would be a major issue for the government and the conservative party (if they were seen as responsible)
The issues with the Irish border are also a concern for the ROI who would be unlikely to agree to a deal with the UK that doesn't resolve the border concerns in a way that would allow some movement between ROI and Northern Island.
As well as the option of having a soft border into the UK through Northern IslandIreland there are a number of other options such as having some form of a border in the Irish sea between the Northern Island the rest of the UK. Which can be resolved at some later point (after all brexit is a long game that no-one really expects to benefit from any time soon) and this can be changed later when the support of the DUP is not so important and as an internal matter that doesn't require the EU to be involved . Equally a soft border is an issue for the EU as well and thus brexiteers would take make the argument that as a matter of mutual concern than some kind of resolution would be possible.
First question: in the hypothesis of a hard Brexit, isn't a soft Irish border a potential backdoor for illegal immigration to the UK? If not what is wrong in the above reasoning?
As you say a hard border is a concert for immigration but there options available down the line that might help reduce the impact of such a border on the rest of the UK.
Second question: assuming that this reasoning is correct, why influential hard-Brexiteers don't campaign for a hard Irish border in order to actually control immigration?
Brexiteers are more concerned with achieving brexit than this specific issue especially as there are options that could protect the rest of the UK from the immigration whilst still avoiding civil unrest. They also require support form the DUP for whatever deal can be made.