Why doesn't blood remain on a mosquito's proboscis in quantities that could spread blood-borne diseases?
A mosquito's proboscis isn't like that of a butterfly, which could easily have nectar clinging to it when it is coiled up; instead, consider that the part of the mosquito's proboscis that enters a blood vessle is probably wiped clean when it is retracts outward through the epithelium. A a dirty needle or razor is many, many times larger than the sucking parts of a mosquito, and a razor is an unlikely vector of HIV.

Epithelial tissue consists of cells very tightly bound together; it has to be to prevent constant invasion by bacteria, fungi, etc. It takes some considerable force or sharpness to penetrate. If you've ever watched someone give you an injection (a vaccine, for example), the needle is never bloody upon withdrawal. You might get a bead of blood from the wound, but the needle itself will be quite clean. The reverse is also true due to essentially the same reasons: the likelihood of introducing a pathogen by dragging it from the surface into the subcutaneous area is so low that there is no absolute need to swab visibly clean skin before giving a subcutaneous injection:


  Although skin that is visibly soiled or dirty must be washed, swabbing the clean skin of a patient before giving an injection is unnecessary. Studies suggest that there is no increased risk of infection when injections were given in the absence of skin preparation. (WHO Bulletin)


HIV is the most serious virus (aside from Ebola, about which much is still unknown) which can be transmitted by needlestick. Sharps injuries - needles, suture needles, scalpels, lancets, etc. - are very common in the medical field, estimated to be much higher than 350,000/year due to underreporting and occurrence in non-hospital settings, e.g. nursing homes, etc. Of course, in the majority of sharps injuries, the sharp wasn't likely to have been used on someone with HIV. Still. HIV is not rare.

Even with a potential half-million exposures a year to sharps, as of December 31, 2013, only 58 confirmed occupational transmissions of HIV and 150 possible transmissions had been reported in the United States. Of these, the vast majority follow hollow needle sticks (88%), where blood drawn from a patient still remains in the bore of the needle which can be reinjected into the unfortunate recipient. If the patient is terminally ill with AIDS, the incidence of transmission by hollow needlestick is increased, meaning the viral load in the blood is important. A solid needle, even a bloody one, presents a very low risk (the risk is, in fact, unknown*.) Getting blood splashed in the eye is a higher risk than a sharp solid (i.e. suture or lancet) needle.

The piercing mouthparts (The mandibles and the maxillae) are solid, akin to a solid sharp. Only the twin tubes (the hypopharynx and the labrum) actually reach blood.

So, given the combined factors of limited survival of HIV on exposed surfaces, the small size of the proboscis (compared to a suturing needle or a lancet, both of which have a higher surface area, yet not known for certain ever to have transmitted HIV), the low probability that the mouth parts have HIV-laden droplets of blood on it, the fact that only the tip of the hypopharynx and labrum encounter blood (the fluid most likely to contain virus) and the extremely low probability that there are enough actual HIV viruses to cause an infection anywhere on the proboscis, I hope this answers your question.

Edited to add: I've focused on HIV here, because it's not known to infect mosquitoes. Mosquitoes are considered by many to be the most dangerous animal in the world. They transmit a significant number of viruses and parasites (like dog heartworm, elephantiasis, dengue fever, equine encephalitis, etc.), but (hypothesized from those diseases that have been studied in mosquitoes) only after being infected by the pathogens themselves. Some die of the infection; some die before the infection spreads through the mosquito's body, some don't harm the mosquito. But transmission is from infected saliva injected into or onto the bite site, not virus particles on the proboscis, as evidenced by the need for adequate incubation time between the bite that takes up the virus to the bite that transmits the virus. If there was sufficient virus on the surface of the proboscis, that incubation time would not be necessary.

Best infection control practices for intradermal, subcutaneous, and intramuscular needle injections
Workbook for Designing, Implementing and Evaluating a Sharps Injury Prevention Program
Occupational HIV infection among health care workers exposed to blood and body fluids in Brazil
Needlesticks: What you must know